Algebra Qual 2023-09

Problem 1

Let $G$ be a group, and let $\mathrm{Aut}(G)$ denote the group of automorphisms of $G$. There is a homomorphism $\gamma :G\to \mathrm{Aut}(G)$ that takes $s\in G$ to the automorphism ${\gamma }_s$ defined by ${\gamma }_s(t)=st{s}^{-1}$.
(a) Prove rigorously, possibly with induction, that is ${\gamma }_s(t)=t^b$, then ${\gamma }_{s^n}(t)=t^{b^n}$.
(b) Suppose $s\in G$ has order 5, and $st{s}^{-1}=t^2$. Find the order of $t$. Justify your answer.

Problem 2

Suppose $G$ is a nonempty finite set that has an associative pairing $G\times G\to G$, written $(x,y)\mapsto x\cdot y$. Suppose this pairing satisfies left and right cancellation: $x\cdot y=x\cdot y^{\prime }$ implies $y=y^{\prime }$, and $x\cdot y=x^{\prime }\cdot y$ implies $x=x^{\prime }$. Prove there exists an element $e$ of $G$ such that for all $x\in G$, $e\cdot x=x\cdot e=x$. Justify your reasoning as carefully as possible.

Problem 3

Let $R_1,\dots ,R_k$ be commutative rings, and set $R=R_1\times \cdots \times R_k$.
(a) Let $I_j\subset R_j$ be ideals, and put $I=I_1\times \cdots \times I_k$. Use the First Isomorphism Theorem to prove that $R/I\simeq R_1/I_1\times \cdots \times R_k/I_k$.
(b) Prove the prime ideals of $R$ have the form $R_1\times \cdots \times R_{j-1}\times P_j\times R_{j+1}\times \cdots \times R_k$ where $P_j\subset R_j$ is a prime ideal for $1\le j\le k$. (Omit the proof that this is an ideal.)

Problem 4

Let $i$ be the imaginary number, let $\mathbf{Z}[i]=\{a+bi\mid a,b\in \mathbf{Z}\}$, a principal ideal domain, and let ${\mathbf{Z}}_2$ be the finite ring of integers modulo 2.
(a) Define a ring homomorphism from $\mathbf{Z}[i]\to {\mathbf{Z}}_2$. You must prove it is a ring homomorphism.
(b) Find, with proof, a generator for the kernel of your ring homomorphism.

Problem 5

Suppose $T:{\mathbf{R}}^n\to {\mathbf{R}}^n$ is a linear transformation with distinct eigenvalues ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m$, and let ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_m$ be corresponding eigenvectors. Prove ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_m$ are linearly independent.