querent

Let $𝑉,𝑊$ be vector spaces over an algebraically closed field $𝑘$, with $dim(𝑉)=6$ and $dim(𝑊)=9$. Suppose $𝑇:𝑉\to 𝑉,𝑆:𝑊\to 𝑊$ are linear transformations whose minimal polynomials are, respectively, ${𝑇}^6=0$ and ${𝑆}^9=0$.

Consider the linear transformation $𝑆\otimes 𝑇:𝑊\otimes 𝑉\to 𝑊\otimes 𝑉$.

(i) What is the minimal polynomial of $𝑆\otimes 𝑇$?

(ii) What is the dimension of $ker(𝑆\otimes 𝑇)$?

(iii) Describe the Jordan normal form of $𝑆\otimes 𝑇$ (i.e., number of blocks, and their sizes).

Hint: You should not need to write down any matrices.

(i) Prove that if a nonzero ideal $𝐼$ in a domain $𝑅$ is free as an $𝑅$-module then $𝐼$ is principal. As an application, for $𝑅=\mathbb{Z}\left[\sqrt{-5}\right]$ prove that neither of the ideals $𝑃=\left(3,1+\sqrt{-5}\right)$ nor $𝑄=\left(3,1-\sqrt{-5}\right)$ is free. Hint: Use norms!

(ii) Prove that $𝑃\cap 𝑄=3𝑅$, and that the addition map $𝑃\oplus 𝑄\to 𝑅$ defined by $(𝑎,𝑏)\mapsto 𝑎+𝑏$ is surjective.

(iii) Deduce that $𝑃\oplus 𝑄\cong {𝑅}^2$ as $𝑅$-modules, so a direct summand of a free module need not be free as an $𝑅$-module!

Suppose $𝐴$ and $𝐵$ are commutative rings containing a field $𝑘$, with $𝐵$ finitely generated over $𝑘$ as a ring. If $𝜙:𝐴\to 𝐵$ is a ring homomorphism with $𝜙{|}_{𝑘}=Id$ and if $𝑄\subset 𝐵$ is a maximal ideal, prove that ${𝜙}^{-1}(𝑄)\subset 𝐴$ is a maximal ideal.

Let $𝐺$ be a finite group and $𝐻$ a subgroup whose index is prime to $𝑝$. Suppose $𝑉$ is a finite-dimensional representation of $𝐺$ over ${𝔽}_{𝑝}$ whose restriction to $𝐻$ is semisimple. Prove that $𝑉$ is semisimple. Hint: Imitate the proof of Maschke's theorem.

Let $𝛼$ be a root of ${𝑥}^7-12$ and $𝜁$ a primitive $7^{\text{th}}$ root of unity, both in $ℂ$.

(a) Explain why the powers ${\{{𝛼}^{𝑗}\}}_{0\le 𝑗\le 6}$ are linearly independent over the field $\mathbb{Q}[𝜁]$.

(b) If $𝛽\in \mathbb{Q}[𝛼]$ has a conjugate of the form ${𝜁}^{𝑖}𝛽$ $(0<𝑖<7)$ in the algebraic closure of $\mathbb{Q}$, explain why $𝛽=𝑐{𝛼}^{𝑗}$ for some rational number $𝑐$ and some $𝑗$ with $0<𝑗<7$.

(c) Show, using the results of the prior parts, that ${𝑥}^7-11$ has no root in $\mathbb{Q}(𝛼)$.

(i) If $𝐴$ is a commutative Noetherian ring, prove the power series ring $𝐴[[𝑥]]$ is Noetherian.

(ii) If $𝐴$ is a commutative Artin ring, that is, if the ideals satisfy the descending chain condition, prove that every prime ideal $𝑃\subset 𝐴$ is maximal and that there are only finitely many prime ideals. Hints: If $𝑥\notin 𝑃$, consider the ideals $({𝑥}^{𝑛})$; consider also finite products of prime ideals.

Let $𝑅$ be a ring and

a complex of free $𝑅$-modules.

(a) Show that this complex is exact (i.e., has vanishing homology) if and only if there exists degree 1 homomorphism $ℎ:{𝐹}_{\ast }\to {𝐹}_{\ast }$ (i.e., a collection of $𝑅$-module homomorphisms ${ℎ}_{𝑗}:{𝐹}_{𝑗}\to {𝐹}_{𝑗+1}$) so that $𝑑ℎ+ℎ𝑑$ is the identity on the complex ${𝐹}_{\ast }$.

(b) In this case, show that $Hom({𝐹}_{\ast },𝑀)$ has vanishing cohomology for any module $𝑀$.

(c) Give counterexamples to both statements if ${𝐹}_{\ast }$ is exact but not free.

Let $𝐾$ be an algebraically closed field and let $𝑉\subset {𝐾}^{𝑛}$, $𝑊\subset {𝐾}^{𝑚}$ be irreducible algebraic sets. Prove that $𝑉\times 𝑊\subset {𝐾}^{𝑛+𝑚}$ is an irreducible algebraic set.

Let $𝑉$ be a nonzero finite-dimensional vector space over an algebraically closed field $𝑘$, and let $𝑇:𝑉\to 𝑉$ be a linear endomorphism.

(a) What does the theorem on Jordan canonical form say about $𝑇$ acting on $𝑉$? Prove it (including the uniqueness aspects) using the structure theorem for finitely generated modules over a PID.

(b) Using the Jordan canonical form, prove that $𝑇$ is diagonalizable if and only if its minimal polynomial has no repeated roots.

(a) Prove that if $𝐺$ is a finite group and $𝐻$ is a proper subgroup, then $𝐺$ is not a union of conjugates of $𝐻$. (Hint: the conjugates all contain the identity.)

(b) Suppose $𝐺$ is a (finite) transitive group of permutations of a finite set $𝑋$ of $𝑛$ objects, $𝑛>1$. Prove that there exists $𝑔\in 𝐺$ with no fixed points in $𝑋$. (Hint: use part (a).)