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(a) Let $𝑅$ be commutative and let $𝑎,𝑏\in 𝑅$. Prove that

(b) Prove that $(5,{𝑥}^2+1)$ is not maximal in $\mathbb{Z}[𝑥]$.

Prove that a commutative ring $𝑅$ satisfies the ascending chain condition for ideals if and only if every ideal of $𝑅$ is finitely generated.

Let $𝑅$ be an integral domain, let $𝑀$ be an $𝑅$-module, and let $𝑁\le 𝑀$. If $𝑁$ and $𝑀/𝑁$ are torsion-free, prove that $𝑀$ is torsion-free.

Let $𝑀$ be a complex $𝑛\times 𝑛$ matrix whose characteristic polynomial has no repeated roots, and suppose all its roots are roots of unity.

(a) Prove that ${𝑀}^{𝑘}={𝐼}_{𝑛}$ for some $𝑘>0$.

(b) Can one necessarily take $𝑘=𝑛$?

(c) Does (a) remain true without separability?

Let $𝐹$ be a field such that every odd-degree polynomial in $𝐹[𝑥]$ has a root in $𝐹$. If $𝐾/𝐹$ is finite and $[𝐾:𝐹]>1$, prove that $[𝐾:𝐹]$ is even.

Let $𝐾/𝐹$ be Galois of degree $25$, and suppose there are two distinct fields strictly between $𝐹$ and $𝐾$. How many strict intermediate fields are there in total?

Consider a finite group $𝐺$ of order ${𝑝}^{𝑛}$, where $𝑝$ is a prime and $𝑛\in {\mathbb{Z}}^{+}$. Prove that every subgroup $𝐻$ of $𝐺$ of index $𝑝$ is a normal subgroup of $𝐺$.

Hint. Consider the action of $𝐺$ on the collection of cosets of $𝐻$ by left multiplication.

In this problem

that is, a root of the polynomial ${𝑥}^2+1\in \mathbb{Q}[𝑥]$.

(a) Is $𝑖\in \mathbb{Q}\left(𝑖\sqrt{2}\right)$? Prove or disprove.

(b) Is $𝑖\in \mathbb{Q}\left(𝑖\sqrt[3]{2}\right)$? Prove or disprove.

Consider a finite group $𝐺$ and a normal subgroup $𝐻$ of $𝐺$. Let $𝑎\in 𝐻$ be such that the conjugacy class $𝐾$ of $𝑎$ in $𝐺$ is contained in $𝐻$. Prove the following.

(a) $𝐾$ is the union of $𝑘$ many conjugacy classes in $𝐻$ of equal size.

(b) $𝑘$ is the index of $𝐻{𝐶}_{𝐺}(𝑎)$ in $𝐺$.

Remark. Recall that ${𝐶}_{𝐺}(𝑎)$ is the centralizer of $𝑎$ in $𝐺$. Also, $𝐻{𝐶}_{𝐺}(𝑎)$ is the product of the two subgroups in $𝐺$.

Denote $𝐺={\text{GL}}_2(ℝ)$, the group of invertible $2\times 2$ real matrices, and

Prove that $𝐻$ is a normal subgroup of $𝐺$ and that $𝐺/𝐻$ is an abelian group.