[ { "id": "Abstract_algebra_0000", "subject": "Abstract_algebra", "topic": "Fields and polynomials", "subtopic": "Polynomials", "level": "3", "keywords": [ "minimal polynomials" ], "problem_v1": "Determine the minimal polynomial $f(x)$ of the following quantities:\n(a) $5+2 i$ over $\\mathbb{R}$, where $i=\\sqrt{-1}$ $f(x)=$ [ANS]\n(b) $5+2 i$ over $\\mathbb{C}$, where $i=\\sqrt{-1}$ $f(x)=$ [ANS]\n(c) $5^{1/4}$ over $\\mathbb{Q}$ $f(x)=$ [ANS]\n(d) $\\sqrt{3}+\\sqrt{5}$ over $\\mathbb{Q}$ $f(x)=$ [ANS]\n(e) $\\sqrt{3}+\\sqrt{5}$ over $\\mathbb{Q}(c)$, where $c=\\sqrt{15}$ $f(x)=$ [ANS]\n(Your answer should be written using $c$, not $\\sqrt{15}$)", "answer_v1": [ "x^2-10*x+29", "x-(5+2*i)", "x^4-5", "x^4-16*x^2+4", "x^2-2*c-8" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Determine the minimal polynomial $f(x)$ of the following quantities:\n(a) $-9+9 i$ over $\\mathbb{R}$, where $i=\\sqrt{-1}$ $f(x)=$ [ANS]\n(b) $-9+9 i$ over $\\mathbb{C}$, where $i=\\sqrt{-1}$ $f(x)=$ [ANS]\n(c) $2^{1/3}$ over $\\mathbb{Q}$ $f(x)=$ [ANS]\n(d) $\\sqrt{13}+\\sqrt{3}$ over $\\mathbb{Q}$ $f(x)=$ [ANS]\n(e) $\\sqrt{13}+\\sqrt{3}$ over $\\mathbb{Q}(c)$, where $c=\\sqrt{39}$ $f(x)=$ [ANS]\n(Your answer should be written using $c$, not $\\sqrt{39}$)", "answer_v2": [ "x^2+18*x+162", "x-(9*i-9)", "x^3-2", "x^4-32*x^2+100", "x^2-2*c-16" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Determine the minimal polynomial $f(x)$ of the following quantities:\n(a) $-4+2 i$ over $\\mathbb{R}$, where $i=\\sqrt{-1}$ $f(x)=$ [ANS]\n(b) $-4+2 i$ over $\\mathbb{C}$, where $i=\\sqrt{-1}$ $f(x)=$ [ANS]\n(c) $3^{1/3}$ over $\\mathbb{Q}$ $f(x)=$ [ANS]\n(d) $\\sqrt{3}+\\sqrt{5}$ over $\\mathbb{Q}$ $f(x)=$ [ANS]\n(e) $\\sqrt{3}+\\sqrt{5}$ over $\\mathbb{Q}(c)$, where $c=\\sqrt{15}$ $f(x)=$ [ANS]\n(Your answer should be written using $c$, not $\\sqrt{15}$)", "answer_v3": [ "x^2+8*x+20", "x-(2*i-4)", "x^3-3", "x^4-16*x^2+4", "x^2-2*c-8" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Abstract_algebra_0001", "subject": "Abstract_algebra", "topic": "Fields and polynomials", "subtopic": "Polynomials", "level": "4", "keywords": [ "quotient fields", "polynomial rings" ], "problem_v1": "Let $t \\in \\mathbb{Q}[x]/(x^2-11)$ be a root of the irreducible polynomial $x^2-11 \\in \\mathbb{Q}[x]$. Express each of the following elements in the form $u+wt$ with $u, w \\in \\mathbb{Q}$. The correct answers may involve fractions.\n(a) $t^5$: [ANS] $+$ [ANS] $t$\n(b) $(6-t)(7+2t)$: [ANS] $+$ [ANS] $t$\n(c) $(7+2t)^2$: [ANS] $+$ [ANS] $t$\n(d) $1/(6-t)$: [ANS] $+$ [ANS] $t$", "answer_v1": [ "0", "121", "20", "5", "93", "28", "0.24", "0.04" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Let $t \\in \\mathbb{Q}[x]/(x^2-3)$ be a root of the irreducible polynomial $x^2-3 \\in \\mathbb{Q}[x]$. Express each of the following elements in the form $u+wt$ with $u, w \\in \\mathbb{Q}$. The correct answers may involve fractions.\n(a) $t^5$: [ANS] $+$ [ANS] $t$\n(b) $(2-t)(1+2t)$: [ANS] $+$ [ANS] $t$\n(c) $(1+2t)^2$: [ANS] $+$ [ANS] $t$\n(d) $1/(2-t)$: [ANS] $+$ [ANS] $t$", "answer_v2": [ "0", "9", "-4", "3", "13", "4", "2", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Let $t \\in \\mathbb{Q}[x]/(x^2-5)$ be a root of the irreducible polynomial $x^2-5 \\in \\mathbb{Q}[x]$. Express each of the following elements in the form $u+wt$ with $u, w \\in \\mathbb{Q}$. The correct answers may involve fractions.\n(a) $t^5$: [ANS] $+$ [ANS] $t$\n(b) $(3-t)(1+2t)$: [ANS] $+$ [ANS] $t$\n(c) $(1+2t)^2$: [ANS] $+$ [ANS] $t$\n(d) $1/(3-t)$: [ANS] $+$ [ANS] $t$", "answer_v3": [ "0", "25", "-7", "5", "21", "4", "0.75", "0.25" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Abstract_algebra_0002", "subject": "Abstract_algebra", "topic": "Fields and polynomials", "subtopic": "Polynomials", "level": "", "keywords": [ "polynomials" ], "problem_v1": "Find a polynomial $f(x)$ of degree 3 over $\\mathbb{Z}_{11}$ such that f(0)=6, \\quad f(7)=7, \\quad f(8)=8 $f(x)=$ [ANS]", "answer_v1": [ "x^3+2*x^2+6" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find a polynomial $f(x)$ of degree 3 over $\\mathbb{Z}_{3}$ such that f(0)=2, \\quad f(1)=1, \\quad f(2)=2 $f(x)=$ [ANS]", "answer_v2": [ "x^3+x^2+2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find a polynomial $f(x)$ of degree 3 over $\\mathbb{Z}_{5}$ such that f(0)=3, \\quad f(2)=2, \\quad f(3)=3 $f(x)=$ [ANS]", "answer_v3": [ "x^3+3*x^2+2*x+3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Abstract_algebra_0003", "subject": "Abstract_algebra", "topic": "Groups", "subtopic": "Group axioms", "level": "3", "keywords": [ "group tables", "center of groups" ], "problem_v1": "The center of a group $G$ is defined to be the set of all elements $x$ in $G$ such that $xy=yx$ for all $y$ in $G$.\nConsider the group whose group table is given as follows ($e$ is the identity element):\n$\\begin{array}{ccccccccccccc}\\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\\\ \\hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\\\ \\hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\\\ \\hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\\\ \\hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\\\ \\hline x6 & x6 & x11 & x10 & x9 & x8 & x7 & x3 & x2 & x1 & e & x5 & x4 \\\\ \\hline x7 & x7 & x6 & x11 & x10 & x9 & x8 & x4 & x3 & x2 & x1 & e & x5 \\\\ \\hline x8 & x8 & x7 & x6 & x11 & x10 & x9 & x5 & x4 & x3 & x2 & x1 & e \\\\ \\hline x9 & x9 & x8 & x7 & x6 & x11 & x10 & e & x5 & x4 & x3 & x2 & x1 \\\\ \\hline x10 & x10 & x9 & x8 & x7 & x6 & x11 & x1 & e & x5 & x4 & x3 & x2 \\\\ \\hline x11 & x11 & x10 & x9 & x8 & x7 & x6 & x2 & x1 & e & x5 & x4 & x3 \\\\ \\hline \\end{array}$\nUsing this group table, determine the elements that lie in the center of $G$ and enter them as a comma-separated list. [ANS]", "answer_v1": [ "(e, x3)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "The center of a group $G$ is defined to be the set of all elements $x$ in $G$ such that $xy=yx$ for all $y$ in $G$.\nConsider the group whose group table is given as follows ($e$ is the identity element):\n$\\begin{array}{ccccccccccccc}\\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline x1 & x1 & e & x3 & x2 & x6 & x7 & x4 & x5 & x11 & x10 & x9 & x8 \\\\ \\hline x2 & x2 & x3 & e & x1 & x7 & x6 & x5 & x4 & x9 & x8 & x11 & x10 \\\\ \\hline x3 & x3 & x2 & x1 & e & x5 & x4 & x7 & x6 & x10 & x11 & x8 & x9 \\\\ \\hline x4 & x4 & x7 & x5 & x6 & x11 & x8 & x10 & x9 & x2 & x1 & x3 & e \\\\ \\hline x5 & x5 & x6 & x4 & x7 & x9 & x10 & x8 & x11 & x1 & x2 & e & x3 \\\\ \\hline x6 & x6 & x5 & x7 & x4 & x8 & x11 & x9 & x10 & x3 & e & x2 & x1 \\\\ \\hline x7 & x7 & x4 & x6 & x5 & x10 & x9 & x11 & x8 & e & x3 & x1 & x2 \\\\ \\hline x8 & x8 & x10 & x11 & x9 & x1 & x3 & x2 & e & x7 & x5 & x4 & x6 \\\\ \\hline x9 & x9 & x11 & x10 & x8 & x3 & x1 & e & x2 & x4 & x6 & x7 & x5 \\\\ \\hline x10 & x10 & x8 & x9 & x11 & x2 & e & x1 & x3 & x6 & x4 & x5 & x7 \\\\ \\hline x11 & x11 & x9 & x8 & x10 & e & x2 & x3 & x1 & x5 & x7 & x6 & x4 \\\\ \\hline \\end{array}$\nUsing this group table, determine the elements that lie in the center of $G$ and enter them as a comma-separated list. [ANS]", "answer_v2": [ "e" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The center of a group $G$ is defined to be the set of all elements $x$ in $G$ such that $xy=yx$ for all $y$ in $G$.\nConsider the group whose group table is given as follows ($e$ is the identity element):\n$\\begin{array}{ccccccccccccc}\\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\\\ \\hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\\\ \\hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\\\ \\hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\\\ \\hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\\\ \\hline x6 & x6 & x7 & x8 & x9 & x10 & x11 & e & x1 & x2 & x3 & x4 & x5 \\\\ \\hline x7 & x7 & x8 & x9 & x10 & x11 & x6 & x1 & x2 & x3 & x4 & x5 & e \\\\ \\hline x8 & x8 & x9 & x10 & x11 & x6 & x7 & x2 & x3 & x4 & x5 & e & x1 \\\\ \\hline x9 & x9 & x10 & x11 & x6 & x7 & x8 & x3 & x4 & x5 & e & x1 & x2 \\\\ \\hline x10 & x10 & x11 & x6 & x7 & x8 & x9 & x4 & x5 & e & x1 & x2 & x3 \\\\ \\hline x11 & x11 & x6 & x7 & x8 & x9 & x10 & x5 & e & x1 & x2 & x3 & x4 \\\\ \\hline \\end{array}$\nUsing this group table, determine the elements that lie in the center of $G$ and enter them as a comma-separated list. [ANS]", "answer_v3": [ "(e, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Abstract_algebra_0004", "subject": "Abstract_algebra", "topic": "Groups", "subtopic": "Group axioms", "level": "3", "keywords": [ "group tables", "order of elements" ], "problem_v1": "The following is the group table of a group whose elements are $\\lbrace e, x1, x2, \\ldots, x11 \\rbrace$, where $e$ is the identity:\n$\\begin{array}{ccccccccccccc}\\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\\\ \\hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\\\ \\hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\\\ \\hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\\\ \\hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\\\ \\hline x6 & x6 & x11 & x10 & x9 & x8 & x7 & x3 & x2 & x1 & e & x5 & x4 \\\\ \\hline x7 & x7 & x6 & x11 & x10 & x9 & x8 & x4 & x3 & x2 & x1 & e & x5 \\\\ \\hline x8 & x8 & x7 & x6 & x11 & x10 & x9 & x5 & x4 & x3 & x2 & x1 & e \\\\ \\hline x9 & x9 & x8 & x7 & x6 & x11 & x10 & e & x5 & x4 & x3 & x2 & x1 \\\\ \\hline x10 & x10 & x9 & x8 & x7 & x6 & x11 & x1 & e & x5 & x4 & x3 & x2 \\\\ \\hline x11 & x11 & x10 & x9 & x8 & x7 & x6 & x2 & x1 & e & x5 & x4 & x3 \\\\ \\hline \\end{array}$\n(a) Express each of the following elements in terms of one element from $\\lbrace e, x1, x2, \\ldots, x11 \\rbrace$.\n$\\begin{array}{cc}\\hline (x7)^2 & [ANS] \\\\ \\hline (x8)^3 & [ANS] \\\\ \\hline (x7)(x8) & [ANS] \\\\ \\hline (x8)(x7) & [ANS] \\\\ \\hline (x7)^{-1} & [ANS] \\\\ \\hline \\end{array}$\n(b) Find all elements $x$ such that $x^3=e$. Enter N if no such element exists. [ANS]\n(c) Find all elements $x$ such that $x^2=x7$. Enter N if no such element exists. [ANS]\n(d) Find all elements $x$ such that $x^3=x8$. Enter N if no such element exists. [ANS]", "answer_v1": [ "x3", "x11", "x2", "x4", "x10", "(e, x2, x4)", "N", "x11" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "UOL", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "The following is the group table of a group whose elements are $\\lbrace e, x1, x2, \\ldots, x11 \\rbrace$, where $e$ is the identity:\n$\\begin{array}{ccccccccccccc}\\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline x1 & x1 & e & x3 & x2 & x6 & x7 & x4 & x5 & x11 & x10 & x9 & x8 \\\\ \\hline x2 & x2 & x3 & e & x1 & x7 & x6 & x5 & x4 & x9 & x8 & x11 & x10 \\\\ \\hline x3 & x3 & x2 & x1 & e & x5 & x4 & x7 & x6 & x10 & x11 & x8 & x9 \\\\ \\hline x4 & x4 & x7 & x5 & x6 & x11 & x8 & x10 & x9 & x2 & x1 & x3 & e \\\\ \\hline x5 & x5 & x6 & x4 & x7 & x9 & x10 & x8 & x11 & x1 & x2 & e & x3 \\\\ \\hline x6 & x6 & x5 & x7 & x4 & x8 & x11 & x9 & x10 & x3 & e & x2 & x1 \\\\ \\hline x7 & x7 & x4 & x6 & x5 & x10 & x9 & x11 & x8 & e & x3 & x1 & x2 \\\\ \\hline x8 & x8 & x10 & x11 & x9 & x1 & x3 & x2 & e & x7 & x5 & x4 & x6 \\\\ \\hline x9 & x9 & x11 & x10 & x8 & x3 & x1 & e & x2 & x4 & x6 & x7 & x5 \\\\ \\hline x10 & x10 & x8 & x9 & x11 & x2 & e & x1 & x3 & x6 & x4 & x5 & x7 \\\\ \\hline x11 & x11 & x9 & x8 & x10 & e & x2 & x3 & x1 & x5 & x7 & x6 & x4 \\\\ \\hline \\end{array}$\n(a) Express each of the following elements in terms of one element from $\\lbrace e, x1, x2, \\ldots, x11 \\rbrace$.\n$\\begin{array}{cc}\\hline (x11)^2 & [ANS] \\\\ \\hline (x2)^3 & [ANS] \\\\ \\hline (x11)(x2) & [ANS] \\\\ \\hline (x2)(x11) & [ANS] \\\\ \\hline (x11)^{-1} & [ANS] \\\\ \\hline \\end{array}$\n(b) Find all elements $x$ such that $x^3=e$. Enter N if no such element exists. [ANS]\n(c) Find all elements $x$ such that $x^2=x11$. Enter N if no such element exists. [ANS]\n(d) Find all elements $x$ such that $x^3=x2$. Enter N if no such element exists. [ANS]", "answer_v2": [ "x4", "x2", "x8", "x10", "x4", "(e, x4, x5, x6, x7, x8, x9, x10, x11)", "x4", "x2" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX", "UOL", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "The following is the group table of a group whose elements are $\\lbrace e, x1, x2, \\ldots, x11 \\rbrace$, where $e$ is the identity:\n$\\begin{array}{ccccccccccccc}\\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\\\ \\hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\\\ \\hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\\\ \\hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\\\ \\hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\\\ \\hline x6 & x6 & x7 & x8 & x9 & x10 & x11 & e & x1 & x2 & x3 & x4 & x5 \\\\ \\hline x7 & x7 & x8 & x9 & x10 & x11 & x6 & x1 & x2 & x3 & x4 & x5 & e \\\\ \\hline x8 & x8 & x9 & x10 & x11 & x6 & x7 & x2 & x3 & x4 & x5 & e & x1 \\\\ \\hline x9 & x9 & x10 & x11 & x6 & x7 & x8 & x3 & x4 & x5 & e & x1 & x2 \\\\ \\hline x10 & x10 & x11 & x6 & x7 & x8 & x9 & x4 & x5 & e & x1 & x2 & x3 \\\\ \\hline x11 & x11 & x6 & x7 & x8 & x9 & x10 & x5 & e & x1 & x2 & x3 & x4 \\\\ \\hline \\end{array}$\n(a) Express each of the following elements in terms of one element from $\\lbrace e, x1, x2, \\ldots, x11 \\rbrace$.\n$\\begin{array}{cc}\\hline (x7)^2 & [ANS] \\\\ \\hline (x3)^3 & [ANS] \\\\ \\hline (x7)(x3) & [ANS] \\\\ \\hline (x3)(x7) & [ANS] \\\\ \\hline (x7)^{-1} & [ANS] \\\\ \\hline \\end{array}$\n(b) Find all elements $x$ such that $x^3=e$. Enter N if no such element exists. [ANS]\n(c) Find all elements $x$ such that $x^2=x7$. Enter N if no such element exists. [ANS]\n(d) Find all elements $x$ such that $x^3=x3$. Enter N if no such element exists. [ANS]", "answer_v3": [ "x2", "x3", "x10", "x10", "x11", "(e, x2, x4)", "N", "(x1, x3, x5)" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "UOL", "EX", "UOL" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Abstract_algebra_0005", "subject": "Abstract_algebra", "topic": "Groups", "subtopic": "Group axioms", "level": "3", "keywords": [ "group tables", "commutativity" ], "problem_v1": "Two elements $x$, $y$ of a group are said to commute with each other if $xy=yx$.\nConsider the group whose group table is given as follows ($e$ is the identity element):\n$\\begin{array}{ccccccccccccc}\\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\\\ \\hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\\\ \\hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\\\ \\hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\\\ \\hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\\\ \\hline x6 & x6 & x11 & x10 & x9 & x8 & x7 & x3 & x2 & x1 & e & x5 & x4 \\\\ \\hline x7 & x7 & x6 & x11 & x10 & x9 & x8 & x4 & x3 & x2 & x1 & e & x5 \\\\ \\hline x8 & x8 & x7 & x6 & x11 & x10 & x9 & x5 & x4 & x3 & x2 & x1 & e \\\\ \\hline x9 & x9 & x8 & x7 & x6 & x11 & x10 & e & x5 & x4 & x3 & x2 & x1 \\\\ \\hline x10 & x10 & x9 & x8 & x7 & x6 & x11 & x1 & e & x5 & x4 & x3 & x2 \\\\ \\hline x11 & x11 & x10 & x9 & x8 & x7 & x6 & x2 & x1 & e & x5 & x4 & x3 \\\\ \\hline \\end{array}$\nUsing this group table, determine the elements that commute with $x7$ and enter them as a comma-separated list. [ANS]", "answer_v1": [ "(e, x3, x7, x10)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Two elements $x$, $y$ of a group are said to commute with each other if $xy=yx$.\nConsider the group whose group table is given as follows ($e$ is the identity element):\n$\\begin{array}{ccccccccccccc}\\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline x1 & x1 & e & x3 & x2 & x6 & x7 & x4 & x5 & x11 & x10 & x9 & x8 \\\\ \\hline x2 & x2 & x3 & e & x1 & x7 & x6 & x5 & x4 & x9 & x8 & x11 & x10 \\\\ \\hline x3 & x3 & x2 & x1 & e & x5 & x4 & x7 & x6 & x10 & x11 & x8 & x9 \\\\ \\hline x4 & x4 & x7 & x5 & x6 & x11 & x8 & x10 & x9 & x2 & x1 & x3 & e \\\\ \\hline x5 & x5 & x6 & x4 & x7 & x9 & x10 & x8 & x11 & x1 & x2 & e & x3 \\\\ \\hline x6 & x6 & x5 & x7 & x4 & x8 & x11 & x9 & x10 & x3 & e & x2 & x1 \\\\ \\hline x7 & x7 & x4 & x6 & x5 & x10 & x9 & x11 & x8 & e & x3 & x1 & x2 \\\\ \\hline x8 & x8 & x10 & x11 & x9 & x1 & x3 & x2 & e & x7 & x5 & x4 & x6 \\\\ \\hline x9 & x9 & x11 & x10 & x8 & x3 & x1 & e & x2 & x4 & x6 & x7 & x5 \\\\ \\hline x10 & x10 & x8 & x9 & x11 & x2 & e & x1 & x3 & x6 & x4 & x5 & x7 \\\\ \\hline x11 & x11 & x9 & x8 & x10 & e & x2 & x3 & x1 & x5 & x7 & x6 & x4 \\\\ \\hline \\end{array}$\nUsing this group table, determine the elements that commute with $x11$ and enter them as a comma-separated list. [ANS]", "answer_v2": [ "(e, x4, x11)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Two elements $x$, $y$ of a group are said to commute with each other if $xy=yx$.\nConsider the group whose group table is given as follows ($e$ is the identity element):\n$\\begin{array}{ccccccccccccc}\\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\\\ \\hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\\\ \\hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\\\ \\hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\\\ \\hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\\\ \\hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\\\ \\hline x6 & x6 & x7 & x8 & x9 & x10 & x11 & e & x1 & x2 & x3 & x4 & x5 \\\\ \\hline x7 & x7 & x8 & x9 & x10 & x11 & x6 & x1 & x2 & x3 & x4 & x5 & e \\\\ \\hline x8 & x8 & x9 & x10 & x11 & x6 & x7 & x2 & x3 & x4 & x5 & e & x1 \\\\ \\hline x9 & x9 & x10 & x11 & x6 & x7 & x8 & x3 & x4 & x5 & e & x1 & x2 \\\\ \\hline x10 & x10 & x11 & x6 & x7 & x8 & x9 & x4 & x5 & e & x1 & x2 & x3 \\\\ \\hline x11 & x11 & x6 & x7 & x8 & x9 & x10 & x5 & e & x1 & x2 & x3 & x4 \\\\ \\hline \\end{array}$\nUsing this group table, determine the elements that commute with $x7$ and enter them as a comma-separated list. [ANS]", "answer_v3": [ "(e, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Abstract_algebra_0006", "subject": "Abstract_algebra", "topic": "Groups", "subtopic": "Subgroups", "level": "3", "keywords": [ "subgroups" ], "problem_v1": "Find all elements of the subgroup $\\langle 12 \\rangle$ in $\\mathbb{Z}_{84}$. [ANS]\nFind all elements of the subgroup $\\langle 9 \\rangle$ in $\\mathbb{Z}_{72}$. [ANS]\nFor both parts, enter your answers as comma-separated lists.", "answer_v1": [ "(0, 12, 24, 36, 48, 60, 72)", "(0, 9, 18, 27, 36, 45, 54, 63)" ], "answer_type_v1": [ "UOL", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "Find all elements of the subgroup $\\langle 6 \\rangle$ in $\\mathbb{Z}_{42}$. [ANS]\nFind all elements of the subgroup $\\langle 12 \\rangle$ in $\\mathbb{Z}_{96}$. [ANS]\nFor both parts, enter your answers as comma-separated lists.", "answer_v2": [ "(0, 6, 12, 18, 24, 30, 36)", "(0, 12, 24, 36, 48, 60, 72, 84)" ], "answer_type_v2": [ "UOL", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "Find all elements of the subgroup $\\langle 8 \\rangle$ in $\\mathbb{Z}_{56}$. [ANS]\nFind all elements of the subgroup $\\langle 11 \\rangle$ in $\\mathbb{Z}_{77}$. [ANS]\nFor both parts, enter your answers as comma-separated lists.", "answer_v3": [ "(0, 8, 16, 24, 32, 40, 48)", "(0, 11, 22, 33, 44, 55, 66)" ], "answer_type_v3": [ "UOL", "UOL" ], "options_v3": [ [], [] ] }, { "id": "Abstract_algebra_0007", "subject": "Abstract_algebra", "topic": "Groups", "subtopic": "Subgroups", "level": "3", "keywords": [ "group tables", "order of elements" ], "problem_v1": "Consider the group whose group table is given as follows ($e$ is the identity element):\n$\\begin{array}{ccccccccccccc}\\hline & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\\\ \\hline e & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\\\ \\hline x_{1} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & e & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} \\\\ \\hline x_{2} & x_{2} & x_{3} & x_{4} & x_{5} & e & x_{1} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} \\\\ \\hline x_{3} & x_{3} & x_{4} & x_{5} & e & x_{1} & x_{2} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} \\\\ \\hline x_{4} & x_{4} & x_{5} & e & x_{1} & x_{2} & x_{3} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} \\\\ \\hline x_{5} & x_{5} & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} \\\\ \\hline x_{6} & x_{6} & x_{11} & x_{10} & x_{9} & x_{8} & x_{7} & x_{3} & x_{2} & x_{1} & e & x_{5} & x_{4} \\\\ \\hline x_{7} & x_{7} & x_{6} & x_{11} & x_{10} & x_{9} & x_{8} & x_{4} & x_{3} & x_{2} & x_{1} & e & x_{5} \\\\ \\hline x_{8} & x_{8} & x_{7} & x_{6} & x_{11} & x_{10} & x_{9} & x_{5} & x_{4} & x_{3} & x_{2} & x_{1} & e \\\\ \\hline x_{9} & x_{9} & x_{8} & x_{7} & x_{6} & x_{11} & x_{10} & e & x_{5} & x_{4} & x_{3} & x_{2} & x_{1} \\\\ \\hline x_{10} & x_{10} & x_{9} & x_{8} & x_{7} & x_{6} & x_{11} & x_{1} & e & x_{5} & x_{4} & x_{3} & x_{2} \\\\ \\hline x_{11} & x_{11} & x_{10} & x_{9} & x_{8} & x_{7} & x_{6} & x_{2} & x_{1} & e & x_{5} & x_{4} & x_{3} \\\\ \\hline \\end{array}$\nDetermine the order of the following elements and complete the table:\n$\\begin{array}{cc}\\hline x & order(x) \\\\ \\hline x_{7} & [ANS] \\\\ \\hline x_{8} & [ANS] \\\\ \\hline x_{7} x_{8} & [ANS] \\\\ \\hline (x_{7})^{2}(x_{8}) & [ANS] \\\\ \\hline (x_{7})^{-1}(x_{8}) & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "4", "4", "3", "4", "6" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Consider the group whose group table is given as follows ($e$ is the identity element):\n$\\begin{array}{ccccccccccccc}\\hline & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\\\ \\hline e & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\\\ \\hline x_{1} & x_{1} & e & x_{3} & x_{2} & x_{6} & x_{7} & x_{4} & x_{5} & x_{11} & x_{10} & x_{9} & x_{8} \\\\ \\hline x_{2} & x_{2} & x_{3} & e & x_{1} & x_{7} & x_{6} & x_{5} & x_{4} & x_{9} & x_{8} & x_{11} & x_{10} \\\\ \\hline x_{3} & x_{3} & x_{2} & x_{1} & e & x_{5} & x_{4} & x_{7} & x_{6} & x_{10} & x_{11} & x_{8} & x_{9} \\\\ \\hline x_{4} & x_{4} & x_{7} & x_{5} & x_{6} & x_{11} & x_{8} & x_{10} & x_{9} & x_{2} & x_{1} & x_{3} & e \\\\ \\hline x_{5} & x_{5} & x_{6} & x_{4} & x_{7} & x_{9} & x_{10} & x_{8} & x_{11} & x_{1} & x_{2} & e & x_{3} \\\\ \\hline x_{6} & x_{6} & x_{5} & x_{7} & x_{4} & x_{8} & x_{11} & x_{9} & x_{10} & x_{3} & e & x_{2} & x_{1} \\\\ \\hline x_{7} & x_{7} & x_{4} & x_{6} & x_{5} & x_{10} & x_{9} & x_{11} & x_{8} & e & x_{3} & x_{1} & x_{2} \\\\ \\hline x_{8} & x_{8} & x_{10} & x_{11} & x_{9} & x_{1} & x_{3} & x_{2} & e & x_{7} & x_{5} & x_{4} & x_{6} \\\\ \\hline x_{9} & x_{9} & x_{11} & x_{10} & x_{8} & x_{3} & x_{1} & e & x_{2} & x_{4} & x_{6} & x_{7} & x_{5} \\\\ \\hline x_{10} & x_{10} & x_{8} & x_{9} & x_{11} & x_{2} & e & x_{1} & x_{3} & x_{6} & x_{4} & x_{5} & x_{7} \\\\ \\hline x_{11} & x_{11} & x_{9} & x_{8} & x_{10} & e & x_{2} & x_{3} & x_{1} & x_{5} & x_{7} & x_{6} & x_{4} \\\\ \\hline \\end{array}$\nDetermine the order of the following elements and complete the table:\n$\\begin{array}{cc}\\hline x & order(x) \\\\ \\hline x_{11} & [ANS] \\\\ \\hline x_{2} & [ANS] \\\\ \\hline x_{11} x_{2} & [ANS] \\\\ \\hline (x_{11})^{2}(x_{2}) & [ANS] \\\\ \\hline (x_{11})^{-1}(x_{2}) & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "3", "2", "3", "3", "3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Consider the group whose group table is given as follows ($e$ is the identity element):\n$\\begin{array}{ccccccccccccc}\\hline & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\\\ \\hline e & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\\\ \\hline x_{1} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & e & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} \\\\ \\hline x_{2} & x_{2} & x_{3} & x_{4} & x_{5} & e & x_{1} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} \\\\ \\hline x_{3} & x_{3} & x_{4} & x_{5} & e & x_{1} & x_{2} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} \\\\ \\hline x_{4} & x_{4} & x_{5} & e & x_{1} & x_{2} & x_{3} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} \\\\ \\hline x_{5} & x_{5} & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} \\\\ \\hline x_{6} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} \\\\ \\hline x_{7} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & e \\\\ \\hline x_{8} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} & x_{2} & x_{3} & x_{4} & x_{5} & e & x_{1} \\\\ \\hline x_{9} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} & x_{3} & x_{4} & x_{5} & e & x_{1} & x_{2} \\\\ \\hline x_{10} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} & x_{4} & x_{5} & e & x_{1} & x_{2} & x_{3} \\\\ \\hline x_{11} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{5} & e & x_{1} & x_{2} & x_{3} & x_{4} \\\\ \\hline \\end{array}$\nDetermine the order of the following elements and complete the table:\n$\\begin{array}{cc}\\hline x & order(x) \\\\ \\hline x_{7} & [ANS] \\\\ \\hline x_{3} & [ANS] \\\\ \\hline x_{7} x_{3} & [ANS] \\\\ \\hline (x_{7})^{2}(x_{3}) & [ANS] \\\\ \\hline (x_{7})^{-1}(x_{3}) & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "6", "2", "6", "6", "6" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Abstract_algebra_0008", "subject": "Abstract_algebra", "topic": "Groups", "subtopic": "Subgroups", "level": "3", "keywords": [ "subgroups", "generators" ], "problem_v1": "Find all elements $x$ in $U(198)$ such that $\\langle x \\rangle$=$\\langle 17 \\rangle$: [ANS]\nAlso find all elements $x$ in $U(189)$ such that $\\langle x \\rangle$=$\\langle 4 \\rangle$: [ANS]\nFor both parts, enter your answers as comma-separated lists.", "answer_v1": [ "(17, 161, 107, 35)", "(4, 16, 67, 79, 130, 142)" ], "answer_type_v1": [ "UOL", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "Find all elements $x$ in $U(164)$ such that $\\langle x \\rangle$=$\\langle 3 \\rangle$: [ANS]\nAlso find all elements $x$ in $U(205)$ such that $\\langle x \\rangle$=$\\langle 3 \\rangle$: [ANS]\nFor both parts, enter your answers as comma-separated lists.", "answer_v2": [ "(3, 27, 79, 55)", "(3, 27, 38, 137)" ], "answer_type_v2": [ "UOL", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "Find all elements $x$ in $U(175)$ such that $\\langle x \\rangle$=$\\langle 6 \\rangle$: [ANS]\nAlso find all elements $x$ in $U(190)$ such that $\\langle x \\rangle$=$\\langle 61 \\rangle$: [ANS]\nFor both parts, enter your answers as comma-separated lists.", "answer_v3": [ "(6, 41, 111, 146)", "(61, 111, 161, 131, 101, 81)" ], "answer_type_v3": [ "UOL", "UOL" ], "options_v3": [ [], [] ] }, { "id": "Abstract_algebra_0009", "subject": "Abstract_algebra", "topic": "Groups", "subtopic": "Subgroups", "level": "4", "keywords": [ "subgroups", "generators" ], "problem_v1": "Find one pair of elements $x,$ $y$ in $U(14)$ such that $\\langle x \\rangle$ and $\\langle y \\rangle$ are proper subgroups of $U(14)$ and that $\\langle x,y \\rangle=U(14)$. Be sure that $x$ $g$ $=k$ [ANS]", "answer_v1": [ "(53, 63, 73, 3, 13, 23, 33, 43)", "(1, -1, k, -k)" ], "answer_type_v1": [ "UOL", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "Let $H$ be a subgroup of $G$. For any element $g$ in $G$. recall that the left coset $gH$ is by definition the subset:\n$gH:=\\lbrace gh: h \\in H \\rbrace$\nFor each of the following triples of $(G, H, g)$, write down the elements of the left coset $gH$ (enter your answer as a comma-separated list):\n(i) $G=\\mathbb{Z}_{56}$, $H=14 \\mathbb{Z}_{56}$, $g=12 \\pmod{56}$ [ANS]\n(ii) $G=$ the quaternion group $Q_8=\\lbrace 1,-1, i,-i, j,-j, k,-k \\rbrace$ $H=\\left$ $g$ $=-i$ [ANS]", "answer_v2": [ "(12, 26, 40, 54)", "(-i, i, k, -k)" ], "answer_type_v2": [ "UOL", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "Let $H$ be a subgroup of $G$. For any element $g$ in $G$. recall that the left coset $gH$ is by definition the subset:\n$gH:=\\lbrace gh: h \\in H \\rbrace$\nFor each of the following triples of $(G, H, g)$, write down the elements of the left coset $gH$ (enter your answer as a comma-separated list):\n(i) $G=\\mathbb{Z}_{55}$, $H=11 \\mathbb{Z}_{55}$, $g=19 \\pmod{55}$ [ANS]\n(ii) $G=$ the quaternion group $Q_8=\\lbrace 1,-1, i,-i, j,-j, k,-k \\rbrace$ $H=\\left$ $g$ $=k$ [ANS]", "answer_v3": [ "(19, 30, 41, 52, 8)", "(k, -k, i, -i)" ], "answer_type_v3": [ "UOL", "UOL" ], "options_v3": [ [], [] ] }, { "id": "Abstract_algebra_0020", "subject": "Abstract_algebra", "topic": "Groups", "subtopic": "Cosets, Lagrange's theorem, and normality", "level": "2", "keywords": [ "cosets", "coset representatives" ], "problem_v1": "Find a complete set of coset representatives of the subgroup $\\langle 19 \\rangle$ in $U(56)$. Enter your answer as a comma separated list; make sure that EACH coset representative you enter\n$\\ast$ is $>0$ and $< 56$, and $\\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $00$ and $< 51$, and $\\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $00$ and $< 35$, and $\\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $00$ and $< 63$, and $\\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $00$ and $< 42$, and $\\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $00$ and $< 52$, and $\\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $00$ and $< 57$. [ANS]", "answer_v1": [ "(4, 6, 8, 10, 12, 0, 2)", "(26, 49, 20, 7, 11, 1)" ], "answer_type_v1": [ "UOL", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "(a) Determine all elements of the coset $1+\\langle 6 \\rangle$ in the group $\\mathbb{Z}_{6}$. Enter your answer as a comma separated list; make sure that each element you enter is $\\geq 0$ and $< 6$. [ANS]\n(b) Determine all elements of the coset $2 \\langle 26 \\rangle$ in the group $U(57)$. Enter your answer as a comma separated list; make sure that each element you enter is $>0$ and $< 57$. [ANS]", "answer_v2": [ "1", "(52, 41, 40, 14, 22, 2)" ], "answer_type_v2": [ "NV", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "(a) Determine all elements of the coset $2+\\langle 6 \\rangle$ in the group $\\mathbb{Z}_{9}$. Enter your answer as a comma separated list; make sure that each element you enter is $\\geq 0$ and $< 9$. [ANS]\n(b) Determine all elements of the coset $1 \\langle 41 \\rangle$ in the group $U(72)$. Enter your answer as a comma separated list; make sure that each element you enter is $>0$ and $< 72$. [ANS]", "answer_v3": [ "(2, 5, 8)", "(41, 25, 17, 49, 65, 1)" ], "answer_type_v3": [ "UOL", "UOL" ], "options_v3": [ [], [] ] }, { "id": "Abstract_algebra_0024", "subject": "Abstract_algebra", "topic": "Groups", "subtopic": "Homomorphisms", "level": "6", "keywords": [ "group homomorphisms", "cyclic groups", "order of groups", "image" ], "problem_v1": "In this problem we determine the number of group homomorphisms\n$f: \\mathbb{Z}_{245} \\rightarrow \\mathbb{Z}_{175}$\nsuch that the image of $f$ has size exactly $7$.\nFirst, since $245 \\equiv 0$ $\\pmod{245}$, we have\n$\\begin{array}{llllllll} 0 & \\equiv & f(0) && \\text{property of homomorphisms} \\\\ & \\equiv & f(245) \\\\ & \\equiv & 245 f(1) \\pmod{175} && \\text{property of homomorphisms.} \\end{array}$ On the other hand, $f(1)$ is an element of $\\mathbb{Z}_{175}$ so $0 \\equiv 175 f(1) \\pmod{175}$.\nThus $\\gcd(245,175) f(1) \\equiv 0$ $\\pmod{175}$ i.e. $35 f(1) \\equiv 0$ $\\pmod{175}$ whence $(\\ast)$ $f(1)=5 u$ for some $u$ in $\\mathbb{Z}_{175}$.\nOn the other hand, the image of $f$ is a subgroup of the cyclic group $\\mathbb{Z}_{175}$. But a cyclic group has at most ONE subgroup of any given order, so if the image of $f$ has size $7$ then the elements of the image of $f$ must be [ANS] (please enter your answer as an ORDERED list)\nCombine this with $(\\ast)$ and we see that in order for the image of $f$ to have size $7$, the choices for $f(1)$ are [ANS] (please enter your answer as an ORDERED list)\nConsequently, the number of such functions $f$ is [ANS]. Please enter your answer as a number.", "answer_v1": [ "(0, 25, 50, 75, 100, 125, 150)", "(25, 50, 75, 100, 125, 150)", "6" ], "answer_type_v1": [ "OL", "OL", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "In this problem we determine the number of group homomorphisms\n$f: \\mathbb{Z}_{28} \\rightarrow \\mathbb{Z}_{98}$\nsuch that the image of $f$ has size exactly $2$.\nFirst, since $28 \\equiv 0$ $\\pmod{28}$, we have\n$\\begin{array}{llllllll} 0 & \\equiv & f(0) && \\text{property of homomorphisms} \\\\ & \\equiv & f(28) \\\\ & \\equiv & 28 f(1) \\pmod{98} && \\text{property of homomorphisms.} \\end{array}$ On the other hand, $f(1)$ is an element of $\\mathbb{Z}_{98}$ so $0 \\equiv 98 f(1) \\pmod{98}$.\nThus $\\gcd(28,98) f(1) \\equiv 0$ $\\pmod{98}$ i.e. $14 f(1) \\equiv 0$ $\\pmod{98}$ whence $(\\ast)$ $f(1)=7 u$ for some $u$ in $\\mathbb{Z}_{98}$.\nOn the other hand, the image of $f$ is a subgroup of the cyclic group $\\mathbb{Z}_{98}$. But a cyclic group has at most ONE subgroup of any given order, so if the image of $f$ has size $2$ then the elements of the image of $f$ must be [ANS] (please enter your answer as an ORDERED list)\nCombine this with $(\\ast)$ and we see that in order for the image of $f$ to have size $2$, the choices for $f(1)$ are [ANS] (please enter your answer as an ORDERED list)\nConsequently, the number of such functions $f$ is [ANS]. Please enter your answer as a number.", "answer_v2": [ "(0, 49)", "49", "1" ], "answer_type_v2": [ "OL", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "In this problem we determine the number of group homomorphisms\n$f: \\mathbb{Z}_{45} \\rightarrow \\mathbb{Z}_{75}$\nsuch that the image of $f$ has size exactly $3$.\nFirst, since $45 \\equiv 0$ $\\pmod{45}$, we have\n$\\begin{array}{llllllll} 0 & \\equiv & f(0) && \\text{property of homomorphisms} \\\\ & \\equiv & f(45) \\\\ & \\equiv & 45 f(1) \\pmod{75} && \\text{property of homomorphisms.} \\end{array}$ On the other hand, $f(1)$ is an element of $\\mathbb{Z}_{75}$ so $0 \\equiv 75 f(1) \\pmod{75}$.\nThus $\\gcd(45,75) f(1) \\equiv 0$ $\\pmod{75}$ i.e. $15 f(1) \\equiv 0$ $\\pmod{75}$ whence $(\\ast)$ $f(1)=5 u$ for some $u$ in $\\mathbb{Z}_{75}$.\nOn the other hand, the image of $f$ is a subgroup of the cyclic group $\\mathbb{Z}_{75}$. But a cyclic group has at most ONE subgroup of any given order, so if the image of $f$ has size $3$ then the elements of the image of $f$ must be [ANS] (please enter your answer as an ORDERED list)\nCombine this with $(\\ast)$ and we see that in order for the image of $f$ to have size $3$, the choices for $f(1)$ are [ANS] (please enter your answer as an ORDERED list)\nConsequently, the number of such functions $f$ is [ANS]. Please enter your answer as a number.", "answer_v3": [ "(0, 25, 50)", "(25, 50)", "2" ], "answer_type_v3": [ "OL", "OL", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Abstract_algebra_0025", "subject": "Abstract_algebra", "topic": "Groups", "subtopic": "Group actions", "level": "6", "keywords": [ "group actions", "orbit-stabilizer theorem" ], "problem_v1": "Let $G$ be a finite group of order $77$ acting on a finite set $S$ of size $11$. What are the possible values for the NUMBER of orbits of this $G$-action? Enter your answer as a comma-separated list. [ANS]", "answer_v1": [ "(1, 5, 11)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Let $G$ be a finite group of order $35$ acting on a finite set $S$ of size $7$. What are the possible values for the NUMBER of orbits of this $G$-action? Enter your answer as a comma-separated list. [ANS]", "answer_v2": [ "(1, 3, 7)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Let $G$ be a finite group of order $15$ acting on a finite set $S$ of size $5$. What are the possible values for the NUMBER of orbits of this $G$-action? Enter your answer as a comma-separated list. [ANS]", "answer_v3": [ "(1, 3, 5)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Abstract_algebra_0026", "subject": "Abstract_algebra", "topic": "Groups", "subtopic": "Group actions", "level": "6", "keywords": [ "group actions", "orbit-stabilizer theorem" ], "problem_v1": "Let $G$ be a finite group of order $35$ acting on a finite set $S$ of size $34$. What are the possible values for the size of the orbit of an element of $S$? Enter your answer as a comma-separated list. [ANS]", "answer_v1": [ "(1, 5, 7)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Let $G$ be a finite group of order $49$ acting on a finite set $S$ of size $17$. What are the possible values for the size of the orbit of an element of $S$? Enter your answer as a comma-separated list. [ANS]", "answer_v2": [ "(1, 7)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Let $G$ be a finite group of order $36$ acting on a finite set $S$ of size $23$. What are the possible values for the size of the orbit of an element of $S$? Enter your answer as a comma-separated list. [ANS]", "answer_v3": [ "(1, 2, 3, 4, 6, 9, 12, 18)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Abstract_algebra_0027", "subject": "Abstract_algebra", "topic": "Groups", "subtopic": "Group actions", "level": "4", "keywords": [ "group actions", "conjugation", "conjugacy classes" ], "problem_v1": "In this problem we determine the conjugacy class of the elements $a^{6} b^0$ and $a^{6} b^1$ in the dihedral group $D_{11}$.\nComplete the table by entering Y or N in each entry:\n$\\begin{array}{ccc}\\hline i & is a^i b^0conjugate to a^{6} b^0? & is a^i b^1conjugate to a^{6} b^1? \\\\ \\hline 0 & [ANS] & [ANS] \\\\ \\hline 1 & [ANS] & [ANS] \\\\ \\hline 2 & [ANS] & [ANS] \\\\ \\hline 3 & [ANS] & [ANS] \\\\ \\hline 4 & [ANS] & [ANS] \\\\ \\hline 5 & [ANS] & [ANS] \\\\ \\hline 6 & [ANS] & [ANS] \\\\ \\hline 7 & [ANS] & [ANS] \\\\ \\hline 8 & [ANS] & [ANS] \\\\ \\hline 9 & [ANS] & [ANS] \\\\ \\hline 10 & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "N", "Y", "N", "Y", "N", "Y", "N", "Y", "N", "Y", "Y", "Y", "Y", "Y", "N", "Y", "N", "Y", "N", "Y", "N", "Y" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "In this problem we determine the conjugacy class of the elements $a^{5} b^0$ and $a^{5} b^1$ in the dihedral group $D_{6}$.\nComplete the table by entering Y or N in each entry:\n$\\begin{array}{ccc}\\hline i & is a^i b^0conjugate to a^{5} b^0? & is a^i b^1conjugate to a^{5} b^1? \\\\ \\hline 0 & [ANS] & [ANS] \\\\ \\hline 1 & [ANS] & [ANS] \\\\ \\hline 2 & [ANS] & [ANS] \\\\ \\hline 3 & [ANS] & [ANS] \\\\ \\hline 4 & [ANS] & [ANS] \\\\ \\hline 5 & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "N", "N", "Y", "Y", "N", "N", "N", "Y", "N", "N", "Y", "Y" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "In this problem we determine the conjugacy class of the elements $a^{5} b^0$ and $a^{5} b^1$ in the dihedral group $D_{8}$.\nComplete the table by entering Y or N in each entry:\n$\\begin{array}{ccc}\\hline i & is a^i b^0conjugate to a^{5} b^0? & is a^i b^1conjugate to a^{5} b^1? \\\\ \\hline 0 & [ANS] & [ANS] \\\\ \\hline 1 & [ANS] & [ANS] \\\\ \\hline 2 & [ANS] & [ANS] \\\\ \\hline 3 & [ANS] & [ANS] \\\\ \\hline 4 & [ANS] & [ANS] \\\\ \\hline 5 & [ANS] & [ANS] \\\\ \\hline 6 & [ANS] & [ANS] \\\\ \\hline 7 & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "N", "N", "N", "Y", "N", "N", "Y", "Y", "N", "N", "Y", "Y", "N", "N", "N", "Y" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Abstract_algebra_0028", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Ring axioms", "level": "2", "keywords": [ "ring axioms" ], "problem_v1": "Let $a=148$ and $b=157$ be elements in the ring $\\mathbb{Z}_{175}$. Evaluate the following expressions. For each one, enter your answer as an integer $0 \\leq n < 175$.\n(a) $a+b=$ [ANS]\n(b) $a-b=$ [ANS]\n(c) $a \\times b=$ [ANS]", "answer_v1": [ "130", "166", "136" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $a=40$ and $b=62$ be elements in the ring $\\mathbb{Z}_{63}$. Evaluate the following expressions. For each one, enter your answer as an integer $0 \\leq n < 63$.\n(a) $a+b=$ [ANS]\n(b) $a-b=$ [ANS]\n(c) $a \\times b=$ [ANS]", "answer_v2": [ "39", "41", "23" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $a=56$ and $b=60$ be elements in the ring $\\mathbb{Z}_{75}$. Evaluate the following expressions. For each one, enter your answer as an integer $0 \\leq n < 75$.\n(a) $a+b=$ [ANS]\n(b) $a-b=$ [ANS]\n(c) $a \\times b=$ [ANS]", "answer_v3": [ "41", "71", "60" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Abstract_algebra_0029", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Ring axioms", "level": "3", "keywords": [ "ring axioms" ], "problem_v1": "For any set $X$, denote by $P^X$ the set of all subsets of $X$ (including the empty set $\\emptyset$ and $X$ itself). This is called the power set of $X$.\nIf $A, B$ are elements of $P^X$, define\n$A+B:=(A-B) \\cup (B-A)$\n$A \\times B:=A \\cap B$\nFACT: $P^X$ together with these two operations forms a commutative ring with a multiplicative identity.\nFor the rest of this exercise, take $X$ to be the set $\\lbrace {1,2,3,4,5,6,7,8,9} \\rbrace$.\n(a) How many elements are there in $P^X$? [ANS]\nNow let $A,B$ be two subsets of $X$ defined as follows:\nA=\\lbrace {1,2,4,5,8,9} \\rbrace\nB=\\lbrace {1,2,7,9} \\rbrace Enter the elements of each set below as a comma-separated list. Input N for the empty set.\n(b) What is the additive inverse of the subset $A$? $\\lbrace$ [ANS] $\\rbrace$\n(c) What is $A+B$? $\\lbrace$ [ANS] $\\rbrace$\n(d) What is $A \\times B$? $\\lbrace$ [ANS] $\\rbrace$", "answer_v1": [ "512", "(1, 2, 4, 5, 8, 9)", "(4, 5, 7, 8)", "(1, 2, 9)" ], "answer_type_v1": [ "NV", "UOL", "UOL", "UOL" ], "options_v1": [ [], [], [], [] ], "problem_v2": "For any set $X$, denote by $P^X$ the set of all subsets of $X$ (including the empty set $\\emptyset$ and $X$ itself). This is called the power set of $X$.\nIf $A, B$ are elements of $P^X$, define\n$A+B:=(A-B) \\cup (B-A)$\n$A \\times B:=A \\cap B$\nFACT: $P^X$ together with these two operations forms a commutative ring with a multiplicative identity.\nFor the rest of this exercise, take $X$ to be the set $\\lbrace {1,2,3,4,5,6} \\rbrace$.\n(a) How many elements are there in $P^X$? [ANS]\nNow let $A,B$ be two subsets of $X$ defined as follows:\nA=\\lbrace {2,3,4,5,6} \\rbrace\nB=\\lbrace {3,4,5} \\rbrace Enter the elements of each set below as a comma-separated list. Input N for the empty set.\n(b) What is the additive inverse of the subset $A$? $\\lbrace$ [ANS] $\\rbrace$\n(c) What is $A+B$? $\\lbrace$ [ANS] $\\rbrace$\n(d) What is $A \\times B$? $\\lbrace$ [ANS] $\\rbrace$", "answer_v2": [ "64", "(2, 3, 4, 5, 6)", "(2, 6)", "(3, 4, 5)" ], "answer_type_v2": [ "NV", "UOL", "UOL", "UOL" ], "options_v2": [ [], [], [], [] ], "problem_v3": "For any set $X$, denote by $P^X$ the set of all subsets of $X$ (including the empty set $\\emptyset$ and $X$ itself). This is called the power set of $X$.\nIf $A, B$ are elements of $P^X$, define\n$A+B:=(A-B) \\cup (B-A)$\n$A \\times B:=A \\cap B$\nFACT: $P^X$ together with these two operations forms a commutative ring with a multiplicative identity.\nFor the rest of this exercise, take $X$ to be the set $\\lbrace {1,2,3,4,5,6,7} \\rbrace$.\n(a) How many elements are there in $P^X$? [ANS]\nNow let $A,B$ be two subsets of $X$ defined as follows:\nA=\\lbrace {1,3,4,6,7} \\rbrace\nB=\\lbrace {1,2,4} \\rbrace Enter the elements of each set below as a comma-separated list. Input N for the empty set.\n(b) What is the additive inverse of the subset $A$? $\\lbrace$ [ANS] $\\rbrace$\n(c) What is $A+B$? $\\lbrace$ [ANS] $\\rbrace$\n(d) What is $A \\times B$? $\\lbrace$ [ANS] $\\rbrace$", "answer_v3": [ "128", "(1, 3, 4, 6, 7)", "(2, 3, 6, 7)", "(1, 4)" ], "answer_type_v3": [ "NV", "UOL", "UOL", "UOL" ], "options_v3": [ [], [], [], [] ] }, { "id": "Abstract_algebra_0030", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Ring axioms", "level": "3", "keywords": [ "ring axioms", "inverse" ], "problem_v1": "(a) Find the multiplicative inverse of $38$ in $\\mathbb{Z}_{39}$. [ANS]\n(b) Find the multiplicative inverse of $30$ in $\\mathbb{Z}_{31}$. [ANS]\n(c) In general, what is the multiplicative inverse of $(n-1)$ in $\\mathbb{Z}_{n}$? [ANS]", "answer_v1": [ "38", "30", "n-1" ], "answer_type_v1": [ "NV", "NV", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "(a) Find the multiplicative inverse of $8$ in $\\mathbb{Z}_{9}$. [ANS]\n(b) Find the multiplicative inverse of $46$ in $\\mathbb{Z}_{47}$. [ANS]\n(c) In general, what is the multiplicative inverse of $(n-1)$ in $\\mathbb{Z}_{n}$? [ANS]", "answer_v2": [ "8", "46", "n-1" ], "answer_type_v2": [ "NV", "NV", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "(a) Find the multiplicative inverse of $19$ in $\\mathbb{Z}_{20}$. [ANS]\n(b) Find the multiplicative inverse of $32$ in $\\mathbb{Z}_{33}$. [ANS]\n(c) In general, what is the multiplicative inverse of $(n-1)$ in $\\mathbb{Z}_{n}$? [ANS]", "answer_v3": [ "19", "32", "n-1" ], "answer_type_v3": [ "NV", "NV", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Abstract_algebra_0031", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Units and zero divisors", "level": "4", "keywords": [ "commutativity", "zero-divisors" ], "problem_v1": "Denote by $R$ the set of all functions from the set $\\lbrace-8, 8, 9, 10 \\rbrace$ to the ring $\\mathbb{Z}_{45}$. FACT: $R$ becomes a ring under the following operations:\n$f+g: a \\mapsto f(a)+g(a), f*g: a \\mapsto f(a)g(a)$ (a) Is $R$ a commutative ring? (Y/N) [ANS]\n(b) How many units are there in $R$? [ANS]\n(c) Give an example of a non-zero $f \\in R$ that is a zero-divisor by filling in the following table:\n$\\begin{array}{cc}\\hline x & f(x) \\\\ \\hline-8 & [ANS] \\\\ \\hline 8 & [ANS] \\\\ \\hline 9 & [ANS] \\\\ \\hline 10 & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "Y", "331776", "(3, 0, 0, 0)" ], "answer_type_v1": [ "TF", "NV", "UOL" ], "options_v1": [ [], [], [] ], "problem_v2": "Denote by $R$ the set of all functions from the set $\\lbrace-10, 2, 8, 10 \\rbrace$ to the ring $\\mathbb{Z}_{30}$. FACT: $R$ becomes a ring under the following operations:\n$f+g: a \\mapsto f(a)+g(a), f*g: a \\mapsto f(a)g(a)$ (a) Is $R$ a commutative ring? (Y/N) [ANS]\n(b) How many units are there in $R$? [ANS]\n(c) Give an example of a non-zero $f \\in R$ that is a zero-divisor by filling in the following table:\n$\\begin{array}{cc}\\hline x & f(x) \\\\ \\hline-10 & [ANS] \\\\ \\hline 2 & [ANS] \\\\ \\hline 8 & [ANS] \\\\ \\hline 10 & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "Y", "4096", "(2, 0, 0, 0)" ], "answer_type_v2": [ "TF", "NV", "UOL" ], "options_v2": [ [], [], [] ], "problem_v3": "Denote by $R$ the set of all functions from the set $\\lbrace-8,-1, 2, 8 \\rbrace$ to the ring $\\mathbb{Z}_{18}$. FACT: $R$ becomes a ring under the following operations:\n$f+g: a \\mapsto f(a)+g(a), f*g: a \\mapsto f(a)g(a)$ (a) Is $R$ a commutative ring? (Y/N) [ANS]\n(b) How many units are there in $R$? [ANS]\n(c) Give an example of a non-zero $f \\in R$ that is a zero-divisor by filling in the following table:\n$\\begin{array}{cc}\\hline x & f(x) \\\\ \\hline-8 & [ANS] \\\\ \\hline-1 & [ANS] \\\\ \\hline 2 & [ANS] \\\\ \\hline 8 & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "TF", "1296", "(2, 0, 0, 0)" ], "answer_type_v3": [ "EX", "NV", "UOL" ], "options_v3": [ [], [], [] ] }, { "id": "Abstract_algebra_0032", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Units and zero divisors", "level": "3", "keywords": [ "characteristic" ], "problem_v1": "(a) Determine the characteristic of the ring $\\mathbb{Z}_{46} \\times \\mathbb{Z}_{36}$. [ANS]\n(b) Determine the characteristic of the ring $\\mathbb{Z}_{38} \\times \\mathbb{Z}_{21}$. [ANS]\n(c) Determine the characteristic of the ring $\\mathbb{Z} \\times \\mathbb{Z}_{35}$. [ANS]", "answer_v1": [ "828", "798", "0" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "(a) Determine the characteristic of the ring $\\mathbb{Z}_{6} \\times \\mathbb{Z}_{56}$. [ANS]\n(b) Determine the characteristic of the ring $\\mathbb{Z}_{10} \\times \\mathbb{Z}_{21}$. [ANS]\n(c) Determine the characteristic of the ring $\\mathbb{Z} \\times \\mathbb{Z}_{57}$. [ANS]", "answer_v2": [ "168", "210", "0" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "(a) Determine the characteristic of the ring $\\mathbb{Z}_{20} \\times \\mathbb{Z}_{18}$. [ANS]\n(b) Determine the characteristic of the ring $\\mathbb{Z}_{34} \\times \\mathbb{Z}_{49}$. [ANS]\n(c) Determine the characteristic of the ring $\\mathbb{Z} \\times \\mathbb{Z}_{55}$. [ANS]", "answer_v3": [ "180", "1666", "0" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Abstract_algebra_0033", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Units and zero divisors", "level": "4", "keywords": [ "characteristic", "integral domains" ], "problem_v1": "For each of the following rings, determine its characteristic and determine if it is an integral domain.\n$\\begin{array}{ccc}\\hline & characteristic & integral domain? (Y/N) \\\\ \\hline \\mathbb{Z} \\times \\mathbb{Z}_{36} & [ANS] & [ANS] \\\\ \\hline \\mathbb{Z}_{44} \\times \\mathbb{Z}_{19} & [ANS] & [ANS] \\\\ \\hline \\mathbb{Z}_{41} \\times \\mathbb{Z}_{41} & [ANS] & [ANS] \\\\ \\hline \\mathbb{Z}[x] & [ANS] & [ANS] \\\\ \\hline \\mathbb{Z}_{38}[x] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "0", "N", "836", "N", "41", "N", "0", "Y", "38", "N" ], "answer_type_v1": [ "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "For each of the following rings, determine its characteristic and determine if it is an integral domain.\n$\\begin{array}{ccc}\\hline & characteristic & integral domain? (Y/N) \\\\ \\hline \\mathbb{Z} \\times \\mathbb{Z}_{56} & [ANS] & [ANS] \\\\ \\hline \\mathbb{Z}_{21} \\times \\mathbb{Z}_{57} & [ANS] & [ANS] \\\\ \\hline \\mathbb{Z}_{3} \\times \\mathbb{Z}_{3} & [ANS] & [ANS] \\\\ \\hline \\mathbb{Z}[x] & [ANS] & [ANS] \\\\ \\hline \\mathbb{Z}_{10}[x] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "0", "N", "399", "N", "3", "N", "0", "Y", "10", "N" ], "answer_type_v2": [ "NV", "EX", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "For each of the following rings, determine its characteristic and determine if it is an integral domain.\n$\\begin{array}{ccc}\\hline & characteristic & integral domain? (Y/N) \\\\ \\hline \\mathbb{Z} \\times \\mathbb{Z}_{37} & [ANS] & [ANS] \\\\ \\hline \\mathbb{Z}_{34} \\times \\mathbb{Z}_{14} & [ANS] & [ANS] \\\\ \\hline \\mathbb{Z}_{13} \\times \\mathbb{Z}_{13} & [ANS] & [ANS] \\\\ \\hline \\mathbb{Z}[x] & [ANS] & [ANS] \\\\ \\hline \\mathbb{Z}_{18}[x] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "0", "N", "238", "N", "13", "N", "0", "Y", "18", "N" ], "answer_type_v3": [ "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV", "TF" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Abstract_algebra_0034", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Units and zero divisors", "level": "2", "keywords": [ "inverse", "zero-divisors" ], "problem_v1": "For each of the following elements of the ring $\\mathbb{Z}_{45}$, determine if it satisfies the following properties. If it is a unit, enter its inverse in the last column; if it is not, enter 0.\n$\\begin{array}{ccc}\\hline & zero-divisor? (Y/N) & inverse \\\\ \\hline 25 & [ANS] & [ANS] \\\\ \\hline 31 & [ANS] & [ANS] \\\\ \\hline 14 & [ANS] & [ANS] \\\\ \\hline 15 & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "Y", "0", "N", "16", "N", "29", "Y", "0" ], "answer_type_v1": [ "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "For each of the following elements of the ring $\\mathbb{Z}_{30}$, determine if it satisfies the following properties. If it is a unit, enter its inverse in the last column; if it is not, enter 0.\n$\\begin{array}{ccc}\\hline & zero-divisor? (Y/N) & inverse \\\\ \\hline 9 & [ANS] & [ANS] \\\\ \\hline 13 & [ANS] & [ANS] \\\\ \\hline 5 & [ANS] & [ANS] \\\\ \\hline 23 & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "Y", "0", "N", "7", "Y", "0", "N", "17" ], "answer_type_v2": [ "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "For each of the following elements of the ring $\\mathbb{Z}_{18}$, determine if it satisfies the following properties. If it is a unit, enter its inverse in the last column; if it is not, enter 0.\n$\\begin{array}{ccc}\\hline & zero-divisor? (Y/N) & inverse \\\\ \\hline 5 & [ANS] & [ANS] \\\\ \\hline 11 & [ANS] & [ANS] \\\\ \\hline 6 & [ANS] & [ANS] \\\\ \\hline 15 & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "N", "11", "N", "5", "Y", "0", "Y", "0" ], "answer_type_v3": [ "TF", "NV", "TF", "NV", "TF", "NV", "TF", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Abstract_algebra_0035", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Units and zero divisors", "level": "3", "keywords": [ "zero-divisors" ], "problem_v1": "For which of the integers $n \\in \\lbrace 34, 59, 73, 25, 30, 50, 58, 38 \\rbrace$ is it true that if $x, y \\in \\mathbb{Z}_{n}$ satisfy $45x=45 y$ then $x=y$? [ANS]", "answer_v1": [ "(73, 34, 58, 59, 38)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "For which of the integers $n \\in \\lbrace 37, 35, 34, 19, 32, 18, 7, 95 \\rbrace$ is it true that if $x, y \\in \\mathbb{Z}_{n}$ satisfy $30x=30 y$ then $x=y$? [ANS]", "answer_v2": [ "(7, 19, 37)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "For which of the integers $n \\in \\lbrace 87, 92, 35, 82, 21, 29, 55, 19 \\rbrace$ is it true that if $x, y \\in \\mathbb{Z}_{n}$ satisfy $18x=18 y$ then $x=y$? [ANS]", "answer_v3": [ "(55, 35, 19, 29)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Abstract_algebra_0036", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Units and zero divisors", "level": "6", "keywords": [ "zero divisors", "Chinese remainder theorem" ], "problem_v1": "Find all integers $n$ in the set $\\lbrace 33, 31, 48, 29, 24, 23, 26, 53, 42 \\rbrace$ such that the ring $\\mathbb{Z}_{n}$ satisifies the following properties:\n(a) $(xy=0) \\Rightarrow (x=0 \\textrm{or} y=0)$ \n(b) $(xy=xz \\textrm{and} x \\neq 0) \\Rightarrow (y=z)$ \n(c) $(x^2=x) \\Rightarrow (x=0 \\textrm{or} x=1)$ [ANS]\nReminder: In the ring $\\mathbb{Z}_{n}$, $0$ means $0 \\pmod{n}$, and equality means congruent modulo $n$. Hint for (c): The Chinese reminder theorem could be useful here.", "answer_v1": [ "(31, 53, 23, 29)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find all integers $n$ in the set $\\lbrace 26, 38, 41, 43, 48, 31, 42 \\rbrace$ such that the ring $\\mathbb{Z}_{n}$ satisifies the following properties:\n(a) $(xy=0) \\Rightarrow (x=0 \\textrm{or} y=0)$ \n(b) $(xy=xz \\textrm{and} x \\neq 0) \\Rightarrow (y=z)$ \n(c) $(x^2=x) \\Rightarrow (x=0 \\textrm{or} x=1)$ [ANS]\nReminder: In the ring $\\mathbb{Z}_{n}$, $0$ means $0 \\pmod{n}$, and equality means congruent modulo $n$. Hint for (c): The Chinese reminder theorem could be useful here.", "answer_v2": [ "(41, 31, 43)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find all integers $n$ in the set $\\lbrace 37, 42, 26, 24, 38, 23, 53, 28, 43, 47, 33 \\rbrace$ such that the ring $\\mathbb{Z}_{n}$ satisifies the following properties:\n(a) $(xy=0) \\Rightarrow (x=0 \\textrm{or} y=0)$ \n(b) $(xy=xz \\textrm{and} x \\neq 0) \\Rightarrow (y=z)$ \n(c) $(x^2=x) \\Rightarrow (x=0 \\textrm{or} x=1)$ [ANS]\nReminder: In the ring $\\mathbb{Z}_{n}$, $0$ means $0 \\pmod{n}$, and equality means congruent modulo $n$. Hint for (c): The Chinese reminder theorem could be useful here.", "answer_v3": [ "(37, 47, 43, 23, 53)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Abstract_algebra_0037", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Ideals and homomorphisms", "level": "6", "keywords": [ "ring homomorphisms" ], "problem_v1": "Determine the number of possible ring homomorphisms for each pair of rings:\n(a) $\\mathbb{Z}_{45} \\rightarrow \\mathbb{Z}_{45}$: [ANS]\n(b) $\\mathbb{Z}_{33} \\rightarrow \\mathbb{Z}_{11}$: [ANS]\n(c) $\\mathbb{Z}_{93} \\rightarrow \\mathbb{Z}_{16}$: [ANS]\nNote: For this problem we do NOT require ring homomorphisms to take the multiplicative identity to the multiplicative identity.", "answer_v1": [ "45", "11", "0" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Determine the number of possible ring homomorphisms for each pair of rings:\n(a) $\\mathbb{Z}_{23} \\rightarrow \\mathbb{Z}_{23}$: [ANS]\n(b) $\\mathbb{Z}_{55} \\rightarrow \\mathbb{Z}_{8}$: [ANS]\n(c) $\\mathbb{Z}_{72} \\rightarrow \\mathbb{Z}_{24}$: [ANS]\nNote: For this problem we do NOT require ring homomorphisms to take the multiplicative identity to the multiplicative identity.", "answer_v2": [ "23", "0", "24" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Determine the number of possible ring homomorphisms for each pair of rings:\n(a) $\\mathbb{Z}_{35} \\rightarrow \\mathbb{Z}_{35}$: [ANS]\n(b) $\\mathbb{Z}_{139} \\rightarrow \\mathbb{Z}_{21}$: [ANS]\n(c) $\\mathbb{Z}_{36} \\rightarrow \\mathbb{Z}_{9}$: [ANS]\nNote: For this problem we do NOT require ring homomorphisms to take the multiplicative identity to the multiplicative identity.", "answer_v3": [ "35", "0", "9" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Abstract_algebra_0038", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Ideals and homomorphisms", "level": "2", "keywords": [ "ideals" ], "problem_v1": "In the ring $\\mathbb{Z}_{88}$, express each of the following ideals in the form $(m)$ for some element $m$ in the ring, where $0 \\leq m < 88$.\n(a) $(51)+(55)$ $($ [ANS] $)$\n(b) $(51)(55)$ $($ [ANS] $)$\n(c) $(51) \\cap (55)$ $($ [ANS] $)$", "answer_v1": [ "1", "77", "77" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "In the ring $\\mathbb{Z}_{54}$, express each of the following ideals in the form $(m)$ for some element $m$ in the ring, where $0 \\leq m < 54$.\n(a) $(49)+(9)$ $($ [ANS] $)$\n(b) $(49)(9)$ $($ [ANS] $)$\n(c) $(49) \\cap (9)$ $($ [ANS] $)$", "answer_v2": [ "1", "9", "9" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "In the ring $\\mathbb{Z}_{65}$, express each of the following ideals in the form $(m)$ for some element $m$ in the ring, where $0 \\leq m < 65$.\n(a) $(39)+(18)$ $($ [ANS] $)$\n(b) $(39)(18)$ $($ [ANS] $)$\n(c) $(39) \\cap (18)$ $($ [ANS] $)$", "answer_v3": [ "3", "52", "52" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Abstract_algebra_0039", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Ideals and homomorphisms", "level": "3", "keywords": [ "maximal ideals" ], "problem_v1": "It is a fact that every ideal of $\\mathbb{Z}_{144}$ is of the form $(b)$ for some element $b$ of $\\mathbb{Z}_{144}$.\n(a) Determine all maximal ideals of $\\mathbb{Z}_{144}$ containing the ideal $(64)$. Enter a generator for each of these ideals. That is, if you think $(64)$ is contained in the maximal ideals $(a)$ and $(b)$, enter $a, b$. [ANS]", "answer_v1": [ "2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "It is a fact that every ideal of $\\mathbb{Z}_{42}$ is of the form $(b)$ for some element $b$ of $\\mathbb{Z}_{42}$.\n(a) Determine all maximal ideals of $\\mathbb{Z}_{42}$ containing the ideal $(36)$. Enter a generator for each of these ideals. That is, if you think $(36)$ is contained in the maximal ideals $(a)$ and $(b)$, enter $a, b$. [ANS]", "answer_v2": [ "(2, 3)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "It is a fact that every ideal of $\\mathbb{Z}_{70}$ is of the form $(b)$ for some element $b$ of $\\mathbb{Z}_{70}$.\n(a) Determine all maximal ideals of $\\mathbb{Z}_{70}$ containing the ideal $(50)$. Enter a generator for each of these ideals. That is, if you think $(50)$ is contained in the maximal ideals $(a)$ and $(b)$, enter $a, b$. [ANS]", "answer_v3": [ "(2, 5)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Abstract_algebra_0040", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Ideals and homomorphisms", "level": "3", "keywords": [ "ideals", "generators" ], "problem_v1": "It is a fact that every ideal of $\\mathbb{Z}_{72}$ is of the form $(b)$ for some element $b$ of $\\mathbb{Z}_{72}$.\n(a) Find all the ideals $I$ of $\\mathbb{Z}_{72}$ that are contained in the ideal $(162)$: $(162) \\subseteq I \\subseteq \\mathbb{Z}_{72}$. In the answer blank below list one generator for each ideal. Separate the generators by commas. [ANS]\n(b) Find all the ideals $J$ of $\\mathbb{Z}_{72}$ that contain the ideal $(162)$: J \\subseteq (162) \\subseteq \\mathbb{Z}_{72}. As in part (a), list one generator for each ideal, separated by commas. [ANS]\nRemember that an ideal contains, and is contained in, itself!", "answer_v1": [ "(0, 18, 36, 54)", "(1, 2, 3, 6, 9, 18)" ], "answer_type_v1": [ "UOL", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "It is a fact that every ideal of $\\mathbb{Z}_{40}$ is of the form $(b)$ for some element $b$ of $\\mathbb{Z}_{40}$.\n(a) Find all the ideals $I$ of $\\mathbb{Z}_{40}$ that are contained in the ideal $(56)$: $(56) \\subseteq I \\subseteq \\mathbb{Z}_{40}$. In the answer blank below list one generator for each ideal. Separate the generators by commas. [ANS]\n(b) Find all the ideals $J$ of $\\mathbb{Z}_{40}$ that contain the ideal $(56)$: J \\subseteq (56) \\subseteq \\mathbb{Z}_{40}. As in part (a), list one generator for each ideal, separated by commas. [ANS]\nRemember that an ideal contains, and is contained in, itself!", "answer_v2": [ "(0, 8, 16, 24, 32)", "(1, 2, 4, 8)" ], "answer_type_v2": [ "UOL", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "It is a fact that every ideal of $\\mathbb{Z}_{108}$ is of the form $(b)$ for some element $b$ of $\\mathbb{Z}_{108}$.\n(a) Find all the ideals $I$ of $\\mathbb{Z}_{108}$ that are contained in the ideal $(60)$: $(60) \\subseteq I \\subseteq \\mathbb{Z}_{108}$. In the answer blank below list one generator for each ideal. Separate the generators by commas. [ANS]\n(b) Find all the ideals $J$ of $\\mathbb{Z}_{108}$ that contain the ideal $(60)$: J \\subseteq (60) \\subseteq \\mathbb{Z}_{108}. As in part (a), list one generator for each ideal, separated by commas. [ANS]\nRemember that an ideal contains, and is contained in, itself!", "answer_v3": [ "(0, 12, 24, 36, 48, 60, 72, 84, 96)", "(1, 2, 3, 4, 6, 12)" ], "answer_type_v3": [ "UOL", "UOL" ], "options_v3": [ [], [] ] }, { "id": "Abstract_algebra_0041", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Ideals and homomorphisms", "level": "2", "keywords": [ "ideals" ], "problem_v1": "(a) Determine all elements in the ideal $(10)$ of $\\mathbb{Z}_{30}$. [ANS]\n(b) Determine all elements in the ideal $(10)+(12)$ of $\\mathbb{Z}_{30}$. [ANS]\n(c) Determine all elements $m$ of $\\mathbb{Z}_{30}$ such that $(10)+(m)$ is a proper ideal of $\\mathbb{Z}_{30}$. [ANS]", "answer_v1": [ "(0, 10, 20)", "(0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28)", "(0, 2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 28)" ], "answer_type_v1": [ "UOL", "UOL", "UOL" ], "options_v1": [ [], [], [] ], "problem_v2": "(a) Determine all elements in the ideal $(14)$ of $\\mathbb{Z}_{28}$. [ANS]\n(b) Determine all elements in the ideal $(14)+(12)$ of $\\mathbb{Z}_{28}$. [ANS]\n(c) Determine all elements $m$ of $\\mathbb{Z}_{28}$ such that $(14)+(m)$ is a proper ideal of $\\mathbb{Z}_{28}$. [ANS]", "answer_v2": [ "(0, 14)", "(0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26)", "(0, 2, 4, 6, 7, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26)" ], "answer_type_v2": [ "UOL", "UOL", "UOL" ], "options_v2": [ [], [], [] ], "problem_v3": "(a) Determine all elements in the ideal $(10)$ of $\\mathbb{Z}_{20}$. [ANS]\n(b) Determine all elements in the ideal $(10)+(8)$ of $\\mathbb{Z}_{20}$. [ANS]\n(c) Determine all elements $m$ of $\\mathbb{Z}_{20}$ such that $(10)+(m)$ is a proper ideal of $\\mathbb{Z}_{20}$. [ANS]", "answer_v3": [ "(0, 10)", "(0, 2, 4, 6, 8, 10, 12, 14, 16, 18)", "(0, 2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18)" ], "answer_type_v3": [ "UOL", "UOL", "UOL" ], "options_v3": [ [], [], [] ] }, { "id": "Abstract_algebra_0042", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Quotient rings and polynomial rings", "level": "5", "keywords": [ "quotient rings", "polynomial rings" ], "problem_v1": "Find all elements $b \\in \\mathbb{Z}_{7}$ such that the quotient ring\n$\\mathbb{Z}_{7} [x]/(x^2+4*x+b)$ is a field. [ANS]", "answer_v1": [ "(1, 5, 6)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find all elements $b \\in \\mathbb{Z}_{2}$ such that the quotient ring\n$\\mathbb{Z}_{2} [x]/(x^2+x+b)$ is a field. [ANS]", "answer_v2": [ "1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find all elements $b \\in \\mathbb{Z}_{3}$ such that the quotient ring\n$\\mathbb{Z}_{3} [x]/(x^2+x+b)$ is a field. [ANS]", "answer_v3": [ "2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Abstract_algebra_0043", "subject": "Abstract_algebra", "topic": "Rings", "subtopic": "Quotient rings and polynomial rings", "level": "3", "keywords": [ "polynomials rings", "associates" ], "problem_v1": "(a) Find all associates of $(7*x^4+7*x^3+8)$ in $\\mathbb{Z}_{12} [x]$. Make sure the coefficients are $\\geq 0$ and $< 12$. [ANS]\n(b) Find all associates of $(1+i)$ in $\\mathbb{Z}[i]$. [ANS]", "answer_v1": [ "(7*x^4+7*x^3+8, 11*x^4+11*x^3+4, x^4+x^3+8, 5*x^4+5*x^3+4)", "(1+i, -1-i, -1+i, 1-i)" ], "answer_type_v1": [ "UOL", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "(a) Find all associates of $(5*x^9+x^3+2)$ in $\\mathbb{Z}_{6} [x]$. Make sure the coefficients are $\\geq 0$ and $< 6$. [ANS]\n(b) Find all associates of $(-6-3i)$ in $\\mathbb{Z}[i]$. [ANS]", "answer_v2": [ "(5*x^9+x^3+2, x^9+5*x^3+4)", "(-6-3i, 6+3i, 3-6i, -3+6i)" ], "answer_type_v2": [ "UOL", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "(a) Find all associates of $(5*x^9+3*x^7+6)$ in $\\mathbb{Z}_{8} [x]$. Make sure the coefficients are $\\geq 0$ and $< 8$. [ANS]\n(b) Find all associates of $(-6-4i)$ in $\\mathbb{Z}[i]$. [ANS]", "answer_v3": [ "(5*x^9+3*x^7+6, 7*x^9+x^7+2, x^9+7*x^7+6, 3*x^9+5*x^7+2)", "(-6-4i, 6+4i, 4-6i, -4+6i)" ], "answer_type_v3": [ "UOL", "UOL" ], "options_v3": [ [], [] ] } ]