MY-ASSIGNMENTEXPERT™可以为您提供 iastate.edu MAT301 Abstract Algebra抽象代数的代写代考和辅导服务！
MAT301课程简介
Prereq: MATH 166 or MATH 166H, MATH 317, and grade of C- or better in MATH 201
Basic properties of integers, divisibility and unique factorization. Polynomial rings over a field. Congruence. Introduction to abstract rings, homomorphisms, ideals. Roots and irreducibility of polynomials. Introduction to groups. Emphasis on proofs.
Prerequisites
Math 166 (Calculus II), Math 317 or 407 (Linear Algebra), and Math 201 (Introduction to Proofs)
A student who has taken Math 207 in lieu of 317 may be prepared for the course. Discuss with the instructor. While the construction of sound proofs will be a central component of the course, a student with no previous experience writing simple proofs may find the course overly challenging.
Prerequisites
Learning Outcomes
Upon completion of this course, students…
Will be familiar with properties of the integers such as prime factorization, divisibility, and congruence
will be able to reason abstractly about mathematical structures
will recognize and comprehend correct proofs of formal statements and be able to formulate proofs clearly and concisely
Learning Objectives
Students will be able to perform computations involving divisibility of integers.
Students will be asked to identify ring-theoretic and group-theoretic properties and identify these properties in familiar rings and groups.
Students will provide proofs to simple assertions of ring- and group-theoretic principles.
MAT301 Abstract Algebra HELP（EXAM HELP， ONLINE TUTOR）
Let $n \geq 1$ be arbitrary. Consider the group $G=\operatorname{Aut}\left(\mathbb{Z}_n\right)$.
(a) Find a group we have seen in this course isomorphic to $G$.
One group we have seen in this course that is isomorphic to $G$ is the group of invertible $n \times n$ matrices with entries in the field $\mathbb{Z}_n$, denoted by $\operatorname{GL}_n(\mathbb{Z}_n)$.
To see why, note that an automorphism of $\mathbb{Z}_n$ is a bijective function $\varphi:\mathbb{Z}_n\to\mathbb{Z}n$ that preserves addition and multiplication modulo $n$. In other words, for all $a,b\in\mathbb{Z}n$, \begin{align*} \varphi(a+b)&\equiv\varphi(a)+\varphi(b)\pmod{n}\ \varphi(ab)&\equiv\varphi(a)\varphi(b)\pmod{n} \end{align*} We can represent $\varphi$ by the $n\times n$ matrix $A=(a{i,j})$ where $a{i,j}=\varphi(i+j)-\varphi(i)-\varphi(j)$ (all arithmetic is done modulo $n$). Note that $A$ is invertible, since its determinant is nonzero in $\mathbb{Z}_n$ if and only if $\varphi$ is a bijection. Moreover, if $\varphi_1,\varphi_2$ are automorphisms of $\mathbb{Z}_n$, then the corresponding matrices $A_1,A_2$
(b) Let $F: G \rightarrow \mathbb{Z}_n$ be the function defined by $F(g)=g(1)$. Explain why $F$ is injective.
To show that $F$ is injective, we need to show that if $F(g_1) = F(g_2)$, then $g_1 = g_2$. So let $g_1, g_2 \in G$ be such that $F(g_1) = F(g_2)$. Then we have $g_1(1) \equiv g_2(1) \pmod{n}$, which implies $g_1(0+1) \equiv g_2(0+1) \pmod{n}$. Since $g_1$ and $g_2$ are automorphisms of $\mathbb{Z}_n$, they preserve addition modulo $n$, so we have $g_1(0)+g_1(1) \equiv g_2(0)+g_2(1) \pmod{n}$. But $g_1(0) \equiv 0 \pmod{n}$ and $g_2(0) \equiv 0 \pmod{n}$ since they are automorphisms, so we can subtract $g_1(0)$ from both sides to get $g_1(1) \equiv g_2(1) \pmod{n}$.
Repeating this argument for $g_1(2)$ and $g_2(2)$, and so on up to $g_1(n-1)$ and $g_2(n-1)$, we get $g_1(k) \equiv g_2(k) \pmod{n}$ for all $k = 0, 1, \ldots, n-1$. But this means that $g_1$ and $g_2$ agree on all elements of $\mathbb{Z}_n$, so they must be equal. Therefore, $F$ is injective.
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