querent

All ring homomorphisms are required to preserve identity.

(a) Describe all ring homomorphisms $\mathbb{Z}\to \mathbb{Z}\times \mathbb{Z}$.

(b) Describe all ring homomorphisms $\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$.

Let $𝑅$ be a ring and $𝑀$ an $𝑅$-module. Prove that $𝑀$ is finitely generated if and only if it is isomorphic to a quotient of a finite-rank free $𝑅$-module.

Does there exist a $14\times 14$ integer matrix of order $49$?

Prove that ${𝑥}^5-2$ is irreducible over $\mathbb{Q}({𝜁}_5)$.

Find the cardinalities of the following subsets of ${𝔽}_{3^6}$:

(a) $\{{𝑥}^5:𝑥\in {𝔽}_{3^6}\}$.

(b) $\{𝑥\in {𝔽}_{3^6}:{𝑥}^3=𝑥\}$.

Suppose $𝐾/𝐹$ is Galois and $Gal(𝐾/𝐹)\cong {𝑄}_8$. Can $𝐾$ be the splitting field of a degree-$4$ polynomial over $𝐹$?

Let $𝐺$ be a group and let

Prove that $𝐻\text{◃}𝐺\times 𝐺$ if and only if $𝐺$ is abelian.

Let $𝐾=𝐹(𝛼)$ be a finite Galois extension, where $𝛼\notin 𝐹$ and $𝛼\ne {𝛼}^{-1}$. Suppose some $𝜎\in Gal(𝐾/𝐹)$ satisfies $𝜎(𝛼)={𝛼}^{-1}$. Prove that

Let $𝑝$ be prime and $𝑘\ge 2$. Prove that no group of order ${𝑝}^{𝑘}(𝑝+1)$ is simple.

Let $𝑅$ be a commutative ring with identity. Assume $𝑅[𝑥]$ is a PID. Prove that $𝑅$ is a field.