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Suppose $𝐴$ is a (commutative) ring, and $𝑀$ is an $𝐴$-module.

(a) If ${𝑀}_{𝔪}=0$ for all maximal ideals $𝔪$ of $𝐴$, must $𝑀=0$? Prove or disprove.

(b) If $𝑀$ is finitely generated and ${𝑀}_{𝔪}/𝔪{𝑀}_{𝔪}=0$ for all maximal ideals $𝔪$ of $𝐴$, prove that $𝑀=0$. Give a counterexample if $𝑀$ is not finitely generated.

Let $𝐸/𝑘$ be a finite-degree extension of fields.

(a) Prove that $Aut(𝐸/𝑘)$ has at most $[𝐸:𝑘]$ elements.

(b) If $𝐸$ is a finite field, prove that $Aut(𝐸/𝑘)$ is cyclic and the norm ${𝐸}^{\times }\to {𝑘}^{\times }$ is surjective.

(c) Give an example of a finite cyclic extension such that the norm is not surjective.

Let $𝐺$ be a finite group, $𝐹$ a field, and $𝑉$ a nonzero finite-dimensional $𝐹$-linear representation of $𝐺$.

(a) Give an example of $𝐺$, $𝐹$, and $𝑉$ such that $𝑉$ does not decompose as a direct sum of irreducible $𝐹$-linear representations of $𝐺$.

(b) Suppose that the order of $𝐺$ is not zero in $𝐹$. Prove that $𝑉$ is a direct sum of irreducible $𝐹$-linear representations of $𝐺$.

Let ${𝐶}_{•}$ be a complex of free abelian groups, with differential lowering degree by 1. Let $𝐴$ be an abelian group.

(a) Construct a short exact sequence

(Hint: use a short free abelian group resolution of $𝐴$.)

(b) Suppose that ${𝐻}_{𝑛}({𝐶}_{•})=0$ except for $𝑛=0$, and ${𝐻}_0({𝐶}_{•})\cong {\mathbb{Z}}_{(5)}\oplus \mathbb{Z}/5\mathbb{Z}$. Here ${\mathbb{Z}}_{(5)}$ is the localization of $\mathbb{Z}$ at the prime $(5)$. Compute the homology groups of ${𝐶}_{•}{\otimes }_{\mathbb{Z}}(\mathbb{Q}/\mathbb{Z})$.

(a) State and prove the Hilbert basis theorem.

(b) Let $𝐴$ be a Noetherian ring, and $𝐽$ an ideal of $𝐴$. Define the ring ${𝐺}_{𝐽}(𝐴)=𝐴\oplus 𝐽\oplus {𝐽}^2\oplus \cdots$, in which the product of ${𝐽}^{𝑚}$ and ${𝐽}^{𝑛}$ is the usual product valued in the direct summand ${𝐽}^{𝑛+𝑚}$. Prove that ${𝐺}_{𝐽}(𝐴)$ is Noetherian.

Let $𝑘$ be a field. Prove that for any $𝑀\in {Mat}_{𝑛}(𝑘)$ ($𝑛\ge 1$), the transpose ${𝑀}^{𝑇}$ of $𝑀$ is conjugate to $𝑀$ in ${Mat}_{𝑛}(𝑘)$ (i.e., ${𝑀}^{𝑇}=𝑔𝑀{𝑔}^{-1}$ for some $𝑔\in {GL}_{𝑛}(𝑘)$).

Let $𝑓:𝐴\to 𝐵$ be a ring homomorphism.

(a) Define what it means to say that $𝐵$ is integral over $𝐴$, and prove this holds when $𝐵$ is finitely generated as an $𝐴$-module.

(b) If $𝐵$ is finitely generated as an $𝐴$-module, prove that $Spec(𝐵)\to Spec(𝐴)$ is a closed map. (Hint: reduce to the case where $𝑓$ is injective.)

Let $𝑝$ be an odd prime. In this question, ${𝜁}_{𝑝}$ denotes a primitive $𝑝$th root of unity.

(a) Describe $Gal(\mathbb{Q}({𝜁}_{𝑝})/\mathbb{Q})$ and determine all primes $𝑝$ such that $\mathbb{Q}({𝜁}_{𝑝})$ contains a subfield $𝐿$ whose Galois group over $\mathbb{Q}$ is isomorphic to $\mathbb{Z}/5\mathbb{Z}$.

(b) Prove that there exists a finite Galois extension $𝐸$ of $\mathbb{Q}$ such that $Gal(𝐸/\mathbb{Q})\cong \mathbb{Z}/5\mathbb{Z}\times \mathbb{Z}/5\mathbb{Z}$ by constructing $𝐸$ as a subfield of an explicit Galois extension $𝐹/\mathbb{Q}$, and explicitly describe the subgroup $Gal(𝐹/𝐸)\subset Gal(𝐹/\mathbb{Q})$.

Let $𝐴$ be a commutative ring, and $𝐺$ a finite group of automorphisms of $𝐴$.

(i) Show that $𝐴$ is integral over the subring ${𝐴}^{𝐺}$ of $𝐺$-invariant elements.

(ii) Let $𝐾$ be a field. If $𝐾$ is algebraically closed, show that a homomorphism ${𝐴}^{𝐺}\to 𝐾$ extends to a homomorphism $𝐴\to 𝐾$. Give (without proof) a counterexample for more general $𝐾$.

(iii) Give an example to show that the extension of (ii) need not be unique.

Let $𝐾$ be a field of finite degree over $\mathbb{Q}$, and ${𝐴}_{𝐾}$ the integral closure of $\mathbb{Z}$ in $𝐾$.

(i) For $𝑥\in 𝐾$, prove $𝑥\in {𝐴}_{𝐾}$ if and only if the subring $\mathbb{Z}[𝑥]\subset 𝐾$ generated by 1 and $𝑥$ is finitely generated as a $\mathbb{Z}$-module.

(ii) Let $𝐾=\mathbb{Q}\left(\sqrt{5}\right)$, determine (with proof) ${𝐴}_{𝐾}$.

(iii) Suppose $[𝐾:\mathbb{Q}]=3$ and there are three prime ideals of ${𝐴}_{𝐾}$ that contain 2. (An example is $𝐾=\mathbb{Q}(𝑎)$ with ${𝑎}^3-{𝑎}^2-2𝑎-8=0$; no need to justify this.) Prove that there are exactly three homomorphisms ${𝐴}_{𝐾}\to {𝔽}_2$. Deduce that ${𝐴}_{𝐾}\ne \mathbb{Z}[𝑦]$ for every $𝑦\in {𝐴}_{𝐾}$.