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Let $𝑅$ be a commutative ring.

(i) Prove any surjective homomorphism of $𝑅$-modules $𝑓:{𝑅}^{𝑛}\to {𝑅}^{𝑛}$ is injective.

(ii) Suppose $𝑅$ is Noetherian and $𝑓:𝑅\to 𝑅$ is a ring homomorphism. Prove again that $𝑓$ is injective if it is surjective. (Hint: consider the kernels of iterates of $𝑓$.)

(iii) Give a counterexample to (ii) if the Noetherian hypothesis is dropped.

(i) Prove that the prime ideals of $\mathbb{Q}[2^{1/3}]{\otimes }_{\mathbb{Q}}\mathbb{Q}[2^{1/3}]$ are principal by exhibiting an explicit generator for each one (written as a sum of elementary tensors).

(ii) Do the same for ${𝔽}_7[𝛼]{\otimes }_{{𝔽}_7}{𝔽}_7[𝛼]$, where ${𝛼}^3=2$.

Let $𝑓:𝐵\to 𝐴$ be a surjective homomorphism of $𝑅$-algebras ($𝑅$ a commutative ring) and let $𝐽=ker(𝑓)$.

(i) For any $𝑅$-algebra ${𝑅}^{\prime }$, let ${𝑓}^{\prime }:{𝐵}^{\prime }\to {𝐴}^{\prime }$ be the ${𝑅}^{\prime }$-algebra homomorphism induced by applying scalar extension ${𝑅}^{\prime }{\otimes }_{𝑅}(\cdot )$. Show that ${𝑓}^{\prime }$ is surjective and that $𝐽{\otimes }_{𝑅}{𝑅}^{\prime }\to {𝐵}^{\prime }$ has image $ker({𝑓}^{\prime })$.

(ii) Prove that $𝐽{\otimes }_{𝑅}{𝑅}^{\prime }\to ker({𝑓}^{\prime })$ is injective (hence an isomorphism) if $𝐴$ is $𝑅$-flat, and give an example with $𝑅=ℂ[𝑥]$ and some $𝐴$ that is not $𝑅$-flat for which this injectivity fails.

(iii) Suppose $𝑅$ and $𝐵$ are local Noetherian, and $𝐴$ is $𝑅$-flat. If the structure map $ℎ:𝑅\to 𝐵$ is local (i.e., $ℎ({𝔪}_{𝑅})\subset {𝔪}_{𝐵}$) and $𝑓mod{𝔪}_{𝑅}$ is an isomorphism, prove that $𝑓$ is an isomorphism. Give a counterexample if $𝐴$ is not assumed to be $𝑅$-flat.

Let $𝐸$ be a field of characteristic zero.

(i) Consider a prime $𝑞$ and an element $𝑏\in {𝐸}^{\times }$ that isn't a $𝑞$th power. Let ${𝐸}^{\prime }=𝐸(𝑎)$ with ${𝑎}^{𝑞}=𝑏$ and ${𝐸}^{\prime }\ne 𝐸$. Show that ${𝑋}^{𝑞}-𝑏$ is reducible over $𝐸$ if and only if $[{𝐸}^{\prime }:𝐸]<𝑞$, and that in such cases ${𝐸}^{\prime }$ contains a primitive $𝑞$th root of unity. Hint: If $1<𝑑=[{𝐸}^{\prime }:𝐸]<𝑞$ then apply the norm ${𝑁}_{{𝐸}^{\prime }/𝐸}$ to the equation ${𝑎}^{𝑞}=𝑏$ to infer that $𝑏$ has a $𝑞$th root in $𝐸$, and compare it to $𝑎$.

(ii) If $𝐾/𝐸$ is a Galois extension of prime degree $𝑝$ and ${𝐸}^{\prime }/𝐸$ is an extension such that $𝐾$ does not admit an $𝐸$-embedding into ${𝐸}^{\prime }$ then show that $𝐾{𝐸}^{\prime }/{𝐸}^{\prime }$ is Galois of degree $𝑝$ and the restriction map $Gal(𝐾{𝐸}^{\prime }/{𝐸}^{\prime })\to Gal(𝐾/𝐸)$ is an isomorphism. (This does not use part (i).)

(iii) Let $𝐸$ be a subfield of $ℝ$ and let $𝐾/𝐸$ be a finite Galois extension of odd degree $>1$. Prove that $𝐾$ cannot be $𝐸$-embedded into a radical tower that is a subfield of $ℝ$. (Hint: Use group theory to reduce to the case when $𝐾/𝐸$ has prime degree $𝑝>2$, and use (i) and (ii) to obtain a contradiction by using that $ℝ$ does not contain nontrivial roots of unity of odd prime order.)

Suppose $𝐴$ is a Noetherian ring.

(i) Prove that the ring of formal power series $𝐴[[𝑥]]$ is also Noetherian.

(ii) Let $𝐼=({𝑎}_1,\dots ,{𝑎}_{𝑛})$ be an ideal of $𝐴$. Let $\widehat{𝐴}$ be the completion of $𝐴$ for the $𝐼$-adic topology. Prove the existence of a surjective ring homomorphism $𝐴[[{𝑥}_1,\dots ,{𝑥}_{𝑛}]]\to \widehat{𝐴}$, and deduce that $\widehat{𝐴}$ is Noetherian.

Let $𝐺$ be a group, $𝑉$ a nonzero finite-dimensional vector space over an algebraically closed field $𝑘$ (allowing $char(𝑘)>0$), and $𝜌:𝐺\to GL(𝑉)$ a representation. Let $𝐸={End}_{𝐺}(𝑉)$ be the algebra of endomorphisms of $𝑉$ commuting with $𝜌(𝑔)$ for all $𝑔\in 𝐺$. Say $𝑉$ is decomposable if $𝑉$ is isomorphic to a direct sum ${𝑉}_1\oplus {𝑉}_2$ of two nonzero $𝐺$-representations.

(i) Prove that $𝑉$ is not decomposable if and only if every element of $𝐸$ is (uniquely) expressible in the form $𝑆+𝑁$, where $𝑆$ is a scalar transformation (i.e. a multiple of the identity) and $𝑁$ is nilpotent. (Hint: Adapt the proof of Schur's lemma, using generalized eigenspaces.)

(ii) Suppose that $𝑉$ is not decomposable. Show that every element of $𝐸$ is either invertible or nilpotent, and prove that the nilpotent elements of $𝐸$ form a two-sided ideal $𝐼$ inside $𝐸$.

(iii) Prove that ${𝐼}^{𝑛}=0$ for some $𝑛\ge 1$ (i.e., ${𝑥}_1\cdots {𝑥}_{𝑛}=0$ with any ${𝑥}_{𝑖}\in 𝐼$).

Let us say that a subgroup $𝐻$ of a group $𝐺$ is a malnormal subgroup if $𝑔𝐻{𝑔}^{-1}\cap 𝐻=\{1\}$ for all $𝑔\in 𝐺-𝐻$.

Let $𝐺$ be a finite group acting transitively on a set $𝑆$. We call $𝐺$ a Frobenius group if no nontrivial element $𝑔\ne 1$ of $𝐺$ fixes more than one element of $𝑆$.

(a) Choose $𝑥\in 𝑆$ and set $𝐻={Stab}_{𝐺}(𝑥)=\{𝑔\in 𝐺|𝑔𝑥=𝑥\}$. Prove that $𝐺$ is a Frobenius group if and only if $𝐻$ is a malnormal subgroup of $𝐺$.

(b) Let $𝐺=\left\{\left(\begin{array}{cc}𝑎& 𝑏\\ 0& 1\end{array}\right)\in {GL}_2({𝔽}_{𝑞})\right\}$ and $𝐻=\left\{\left(\begin{array}{cc}𝑎& 0\\ 0& 1\end{array}\right)\in {GL}_2({𝔽}_{𝑞})\right\}$. Prove that $𝐻$ is a malnormal subgroup of $𝐺$. (Hint: Let $𝑆={𝔽}_{𝑞}$.)

Let $𝐺$ be a finite group. By a $𝐺$-module we mean a $ℂ[𝐺]$-module of finite $ℂ$-dimension.

(a) Let ${𝑉}_1,\cdots ,{𝑉}_{ℎ}$ be pairwise non-isomorphic irreducible $𝐺$-modules. Let

be a module that contains ${𝑉}_{𝑖}$ exactly ${𝑑}_{𝑖}$ times. Describe the structure of the ring ${End}_{𝐺}(𝑉)$. (Briefly justify your answer.)

(b) Suppose that $𝑉$ is a nonzero $𝐺$-module that is not irreducible. Prove that the ring ${End}_{𝐺}(𝑉\otimes 𝑉)$ is not commutative.

Let $𝐾$ be a field.

(a) Compute

(b) Let $𝑉$ be an $ℝ$-vector space of dimension $𝑛$, and let $𝐴:𝑉\to 𝑉$ be a linear transformation. Prove that the following are equivalent:

(i) ${𝐴}^{𝑛}=0$.

(ii) For every $𝜀>0$ there is a basis of $𝑉$ such that every entry of the matrix of $𝐴$ has absolute value $<𝜀$.

(c) Explain how find representatives of the equivalence classes of nilpotent $𝑛\times 𝑛$ matrices over $𝐾$ under conjugation by $GL(𝑛,𝐾)$. (Hint: rational canonical form.)

(a) Prove that if $𝛾\in GL(3,\mathbb{Z})$ has finite order $𝑛$ then $𝑛\le 6$. (Hint: What can you say about the eigenvalues of $𝛾$?)

(b) Exhibit $𝛾\in GL(3,\mathbb{Z})$ of order 6.