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Let $𝐺$ be a non-abelian finite group that is simple.

(i) Give, with proof, an example of such a group.

(ii) Given that $𝐺$ has even order (Feit–Thompson), show that there are no irreducible representations $𝜌:𝐺\to {GL}_2(ℂ)$. (Hint: consider $det𝜌$ and involutions in $𝐺$.)

Let $𝐾$ be an algebraically closed subfield of $ℂ$, and $𝐽\subset 𝐾[{𝑥}_1,\dots ,{𝑥}_{𝑛}]$ an ideal. Let $𝑍$ be the vanishing locus of $𝐽$ in ${ℂ}^{𝑛}$.

(i) Suppose that a polynomial $𝑝\in ℂ[{𝑥}_1,\dots ,{𝑥}_{𝑛}]$ vanishes on $𝑍\cap {𝐾}^{𝑛}$. Show that $𝑝$ vanishes on $𝑍$. (Hint: explain why vanishing on $𝑍\cap {𝐾}^{𝑛}$ is a possibly infinite system of homogeneous linear equations in the coefficients of $𝑝$; deduce via linear algebra that $𝑝=\sum {𝑐}_{𝑖}{𝑝}_{𝑖}$ for ${𝑐}_{𝑖}\in ℂ$ and polynomials ${𝑝}_{𝑖}\in 𝐾[{𝑥}_1,\dots ,{𝑥}_{𝑛}]$ that vanish on $𝑍\cap {𝐾}^{𝑛}$.)

(ii) Prove $𝑍$ is the closure of $𝑍\cap {𝐾}^{𝑛}$ for the Zariski topology on ${ℂ}^{𝑛}$.

(iii) Prove the inclusion ${𝐾}^{𝑛}\to {ℂ}^{𝑛}$ is continuous for the Zariski topologies.

Let $𝐴$ be a noetherian ring and $0\to {𝑀}^{\prime }\to 𝑀\stackrel{𝑓}{\to }{𝑀}^{″}\to 0$ a short exact sequence of $𝐴$-modules with $𝑀$ finitely generated.

(i) If $𝐴$ is local with maximal ideal $𝔪$ and ${𝑀}^{″}$ is $𝐴$-flat, show that $𝑓$ is an isomorphism if the induced map $𝑀/𝔪𝑀\to {𝑀}^{″}/𝔪{𝑀}^{″}$ is an isomorphism and give a counterexample without $𝐴$-flatness.

(ii) If $𝑁$ is a finitely generated and flat $𝐴$-module then show that ${𝑁}_{𝑃}$ is ${𝐴}_{𝑃}$-free for every prime ideal $𝑃$ of $𝐴$. (Hint: Choose an ${𝐴}_{𝑃}/𝑃{𝐴}_{𝑃}$-basis of ${𝑁}_{𝑃}/𝑃{𝑁}_{𝑃}$ and use (i).)

(iii) For $𝑛>0$ and $0\le 𝑖\le 𝑛$, let ${𝐹}^{𝑖}\subset {\wedge }^{𝑛}(𝑀)$ be the submodule spanned by the wedge products ${𝑚}_1\wedge \cdots \wedge {𝑚}_{𝑛}$ in which at least $𝑖$ of the ${𝑚}_{𝑗}$'s lie in ${𝑀}^{\prime }$, and let ${𝐹}^{𝑛+1}=0$. Show there exists a unique $𝐴$-linear map

satisfying

and (using (ii)) prove this is an isomorphism when ${𝑀}^{\prime }$, $𝑀$, and ${𝑀}^{″}$ are $𝐴$-flat. (You may use without proof that exterior powers of modules commute with localization.)

(i) Let $𝐴\subset 𝐵$ be nonzero commutative rings, where $𝐵$ is integral over $𝐴$. Prove that $dim𝐴=dim𝐵$ (clearly stating any theorem that you use).

(ii) State the Noether normalization theorem (for finitely generated domains over an infinite field) and sketch a proof.

(iii) Let $𝐾/𝑘$ be an arbitrary extension of infinite fields, and let $𝐵$ be a finitely generated $𝑘$-algebra that is a domain. Granting that $dim(𝐹[{𝑥}_1,\dots ,{𝑥}_{𝑑}])=𝑑$ for any field $𝐹$ and integer $𝑑>0$, prove that $dim(𝐵{\otimes }_{𝑘}𝐾)=dim(𝐵)$. (Do not assume that $𝐵{\otimes }_{𝑘}𝐾$ is a domain. One point extra credit for handling the case when $𝑘$ is finite.)

Let $𝑅=\mathbb{Z}[𝑡]$ and make $\mathbb{Z}/𝑛\mathbb{Z}$ an $𝑅$-module by letting $𝑡$ act as the identity.

(i) Give an explicit finite resolution of $\mathbb{Z}/2\mathbb{Z}$ by projective $𝑅$-modules.

(ii) Compute ${Ext}_{𝑅}^{𝑖}(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/4\mathbb{Z})$ for every $𝑖\ge 0$.

Let $𝐺={GL}_2(\mathbb{Z}/9\mathbb{Z})$.

(i) Compute the order of $𝐺$.

(ii) Prove $𝑔\in 𝐺$ has 3-power order if and only if its image in ${GL}_2(\mathbb{Z}/3\mathbb{Z})$ does.

(iii) Show that a Sylow 2-subgroup of $𝐺$ is isomorphic to ${𝔽}_9^{\times }⋊(\mathbb{Z}/2\mathbb{Z})$ where the nontrivial element $1\in \mathbb{Z}/2\mathbb{Z}$ acts on ${𝔽}_9^{\times }$ via $𝑥\mapsto {𝑥}^3$.

(i) Prove that ${𝑋}^4-6\in \mathbb{Q}[𝑋]$ is irreducible, and if $𝑎$ denotes a root then show that $-1$ is not a square in $\mathbb{Q}(𝑎)$ and that the Galois closure $𝐾$ of $\mathbb{Q}(𝑎)$ over $\mathbb{Q}$ coincides with $\mathbb{Q}(𝑎,𝑖)$ with ${𝑖}^2=-1$.

(ii) Determine with proof the group $Gal(𝐾/\mathbb{Q})$, and use this to show that $\mathbb{Q}\left(\sqrt{6}\right)$ is the only proper subfield of $\mathbb{Q}(𝑎)$ strictly containing $\mathbb{Q}$.

(iii) Let $𝑝$ be a prime such that ${𝑋}^4-6\in {𝔽}_{𝑝}[𝑋]$ is irreducible, and let $𝑎$ denote a root in an extension of ${𝔽}_{𝑝}$. Show 13 is such a prime, that ${𝔽}_{𝑝}(𝑎)$ is Galois over ${𝔽}_{𝑝}$ of degree 4, and that ${𝑎}^{{𝑝}^2}=-𝑎$.

Let $𝑉$ be a $ℂ$-vector space of even dimension $2𝑛$ and $𝐵:𝑉\times 𝑉\to ℂ$ a symmetric nondegenerate bilinear form. Let ${𝑂}_{𝐵}$ be the subgroup of $GL(𝑉)$ consisting of those automorphisms that preserve $𝐵$; i.e.,

A subspace $𝑊\subset 𝑉$ is isotropic (with respect to $𝐵$) if $𝐵(𝑤,{𝑤}^{\prime })=0$ for all $𝑤,{𝑤}^{\prime }\in 𝑊$.

(i) Prove that an isotropic subspace $𝑊$ is maximal (among isotropic subspaces) if and only if $dim𝑊=𝑛$. (Hint: consider the induced bilinear form on ${𝑊}^{⟂}/𝑊$.)

(ii) Let $𝑊$ be maximal isotropic and suppose that $𝑔\in {𝑂}_{𝐵}$ satisfies $𝑔𝑊=𝑊$. Prove that $det𝑔=1$. (Hint: extend a basis ${𝑤}_1,\dots ,{𝑤}_{𝑛}$ for $𝑊$ to a basis ${𝑤}_1,\dots ,{𝑤}_{𝑛},{𝑣}_1,\dots ,{𝑣}_{𝑛}$ for $𝑉$ such that $𝐵({𝑣}_{𝑖},{𝑤}_{𝑗})$ equals 1 if $𝑖=𝑗$ and equals 0 otherwise. Show the matrix of $𝑔$ with respect to this basis is "block triangular" with $𝑛\times 𝑛$ blocks.)

(iii) For the action $𝑊\mapsto 𝑔𝑊$ of ${𝑂}_{𝐵}$ on the set of maximal isotropic subspaces, show that the determinant-1 subgroup ${SO}_{𝐵}≔{𝑂}_{𝐵}\cap SL(𝑉)$ does not act transitively.

(a) Prove that every group of order $15$ is cyclic.

(b) Is every group of order $21$ cyclic?

Give an example or explain that none exists.

(a) A UFD that is not a Euclidean domain.

(b) A positive integer $𝑛$ such that $\mathbb{Q}({𝜁}_{𝑛})$ contains two distinct quadratic extensions of $\mathbb{Q}$.