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(a) For a commutative ring $𝑅$ and multiplicative set $𝑆\subset 𝑅$, show the functor $𝑀⇝{𝑆}^{-1}𝑀$ on $𝑅$-modules is exact.

(b) Let $𝑘$ be an algebraically closed field and $𝑓,𝑔\in 𝑘[𝑥,𝑦]$ be non-constant with no common irreducible factor. For $𝐴=𝑘[𝑥,𝑦]/(𝑓,𝑔)$, prove ${dim}_{𝑘}𝐴<\infty$ and that there are only finitely many maximal ideals ${𝔪}_1,\dots ,{𝔪}_{𝑛}$ of $𝑘[𝑥,𝑦]$ containing $𝑓$ and $𝑔$. Also show that

Suppose $𝑝$ is a prime number. How many conjugacy classes does the group ${GL}_3(\mathbb{Z}/𝑝\mathbb{Z})$ have?

Suppose that $𝐿/\mathbb{Q}$ is a finite Galois extension and let $𝐵$ denote the integral closure of $\mathbb{Z}$ in $𝐿$, so that $𝐵$ is a Dedekind domain. Also suppose that $𝑝\in \mathbb{Z}$ is a prime number and factor $𝑝𝐵={\prod }_{𝑖=1}^{𝑔}{𝔭}_{𝑖}^{{𝑒}_{𝑖}}$ with the ${𝔭}_{𝑖}$ distinct prime ideals of $𝐵$ and with ${𝑒}_{𝑖}\in {\mathbb{Z}}_{>0}$.

(a) Show that if $𝔭$ is a prime ideal of $𝐵$, then $𝑝\in 𝔭$ if and only if $𝔭={𝔭}_{𝑖}$ for some $𝑖$. In this case show also that $𝔭\cap \mathbb{Q}=𝑝\mathbb{Z}$.

(b) Show that we can find ${𝜛}_{𝑖}\in {𝔭}_{𝑖}-{\bigcup }_{𝑗\ne 𝑖}{𝔭}_{𝑗}$. Show moreover that ${𝑁}_{𝐿/\mathbb{Q}}{𝜛}_{𝑖}\in 𝑝\mathbb{Z}$.

(c) If $𝑖\ne 𝑗$ show that there is $𝜎\in Gal(𝐿/\mathbb{Q})$ with $𝜎{𝜛}_{𝑖}\in {𝔭}_{𝑗}$. Deduce that $𝜎{𝔭}_{𝑖}={𝔭}_{𝑗}$.

(d) Show that ${𝑒}_{𝑖}$ is independent of $𝑖$.

Let $𝐴$ be a principal ideal domain with fraction field $𝐾$, and $𝐿$ a nonzero finite free $𝐴$-module equipped with a bilinear $𝐵:𝐿\times 𝐿\to 𝐴$ such that the scalar extension ${𝐵}_{𝐾}$ is non-degenerate over $𝐾$.

(a) Show the set $\{𝐵(\ell ,{\ell }^{\prime })|\ell ,{\ell }^{\prime }\in 𝐿\}$ coincides with a nonzero ideal $(𝑑)$ of $𝐴$. Hint: Consider the map $𝐿\to {𝐿}^{\ast }$ sending $\ell \in 𝐿$ to the linear functional $𝐵(\ell ,\cdot )$.

(b) Assume $𝐵$ is alternating. If $𝐵(\ell ,{\ell }^{\prime })=𝑑$ for $𝑑$ as in (a), prove that $\ell$ and ${\ell }^{\prime }$ are linearly independent and that $𝐴\ell \oplus 𝐴{\ell }^{\prime }$ is a direct summand of $𝐿$.

Suppose that $𝐿/𝐾$ is a finite extension of fields of characteristic 0, that $-1$ has a square root in $𝐾$ and that $𝐿$ is algebraically closed. In this question you will be asked to prove by contradiction that $𝐿=𝐾$.

(a) Show that if $𝐿\ne 𝐾$, then we can find an intermediate field $𝐿\supset 𝑀\supset 𝐾$ with $[𝐿:𝑀]=𝑝$ a prime.

(b) Show that $𝑀$ contains a primitive ${𝑝}^{\text{th}}$ root of unity, and deduce that $𝐿=𝑀(𝛽)$, where ${𝛽}^{𝑝}=𝛼\in 𝑀$. What is the minimum polynomial of $𝛽$ over $𝑀$? Show moreover that ${𝑁}_{𝐿/𝑀}(𝛽)={(-1)}^{𝑝-1}𝛼$.

(c) Show that $𝛽={𝛾}^{𝑝}$ for some $𝛾\in 𝐿$ and conclude that $𝛼$ is a ${𝑝}^{\text{th}}$ power in $𝑀$ and $𝐿=𝑀$, a contradiction. Hint: take norms, and remember that $-1$ is a square in $𝐾$.

A subring $𝐴$ of a field $𝐾$ is said to be a valuation ring if for every nonzero $𝑥\in 𝐾$, either $𝑥\in 𝐴$ or $1/𝑥\in 𝐴$. Suppose $𝐴\subset 𝐾$ is a valuation ring.

(a) Prove that $𝐴$ is a local ring.

(b) Prove that $𝐴$ is integrally closed.

Let $𝐺$ be a finite group and $𝑉,𝑊$ a pair of $𝐺$-modules, by which we mean finite-dimensional $ℂ$-linear representations of $𝐺$ or equivalently modules over the group algebra $ℂ[𝐺]$. We will denote by $Hom(𝑉,𝑊)$ all $ℂ$-linear transformations $𝑉\to 𝑊$, and by ${Hom}_{𝐺}(𝑉,𝑊)$ the space of those that are $𝐺$-module homomorphisms $𝑓$; i.e. $ℂ$-linear maps that satisfy $𝑓(𝑔𝑣)=𝑔𝑓(𝑣)$. We will denote these spaces $End(𝑉)$ and ${End}_{𝐺}(𝑉)$ if $𝑉=𝑊$. If $𝑉$ is a $𝐺$-module let ${𝑉}^{𝐺}=\{𝑣\in 𝑉|𝑔𝑣=𝑣\text{for all}𝑔\in 𝐺\}$ be the space of $𝐺$-invariants.

(a) Show that $Hom(𝑉,𝑊)$ is itself a $𝐺$-module with the action $(𝑔𝑓)(𝑣)=𝑔𝑓({𝑔}^{-1}𝑣)$, and that the space $Hom{(𝑉,𝑊)}^{𝐺}$ of $𝐺$-invariants is ${Hom}_{𝐺}(𝑉,𝑊)$.

(b) Remind us how the dual space ${𝑉}^{\ast }$ of a $𝐺$-module is a $𝐺$-module, and show that $Hom({𝑉}_1,{𝑉}_2)\cong {𝑉}_1^{\ast }\otimes {𝑉}_2$ as $𝐺$-modules.

(c) Prove that

Hint: First show that $Hom(Hom({𝑉}_1,{𝑉}_2),Hom({𝑊}_1,{𝑊}_2))\cong Hom({𝑉}_2\otimes {𝑊}_1,{𝑉}_1\otimes {𝑊}_2)$.

Let $𝑝$ be a prime number and let ${𝔽}_{𝑝}$ be the finite field of $𝑝$ elements. Let $𝐺={GL}_2({𝔽}_{𝑝})/𝑍$, where ${GL}_2({𝔽}_{𝑝})$ is the group invertible $2\times 2$ matrices with coefficients in ${𝔽}_{𝑝}$, and $𝑍$ is the subgroup of scalar matrices. Note that the natural linear action of ${GL}_2({𝔽}_{𝑝})$ on ${𝔽}_{𝑝}^2$ induces an action of $𝐺$ on the set $𝑷$ of 1-dimensional subspaces in ${𝔽}_{𝑝}^2$.

(a) Show that $𝐺$ acts faithfully and transitively on the set $𝑷$. Using this fact to show that $|𝐺|=𝑝({𝑝}^2-1)$.

(b) Let $𝐵\subset 𝐺$ be a subgroup of order $(𝑝-1)𝑝$. Show that $𝐵$ is the stabilizer of a line $\ell \in 𝑷$, and therefore any two such subgroups are conjugate.

(c) Let $𝐻\subset 𝐺$ be a cyclic subgroup of order $𝑝+1$. Show that $𝐻$ acts simply-transitively on the set $𝑷$.

How many elements of order $7$ are there in a simple group of order $168$?

For each of the questions below, say whether the statement is True or False and give a brief justification.

(a) Let $𝐹\subseteq 𝐾\subseteq 𝐿$ be field extensions for which $𝐾/𝐹$ is Galois and $𝐿/𝐾$ is Galois. Then $𝐿/𝐹$ is also Galois.

(b) Let $𝑅$ be a commutative ring with unity and let $𝑀$ be a finitely generated $𝑅$-module. Let $𝑁$ be a submodule of $𝑀$. Then $𝑁$ is a finitely generated $𝑅$-module.