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Assume $𝐹$ is a field and $𝑥,𝑦$ are variables. Prove that the rings

and

are not isomorphic.

Remark. Recall that if $𝑅$ is a ring and $𝑟\in 𝑅$, then $(𝑟)$ is the ideal of $𝑅$ generated by $𝑟$.

An element $𝑥$ of a ring is nilpotent if ${𝑥}^{𝑛}=0$ for some integer $𝑛\ge 1$. Let $𝑅$ be a commutative ring, and let

Prove that $𝑁$ is an ideal of $𝑅$ and that $𝑅/𝑁$ has no nonzero nilpotent elements.

Suppose that $𝐸$ and $𝐹$ are two fields whose characteristics satisfy

and $𝑝\ne 𝑞$. Prove that the abelian group

is zero.

Consider an $𝑛\times 𝑛$ matrix $𝐴$ with entries in $\mathbb{Q}$ which satisfies

where $𝐼$ is the identity matrix. Prove the following.

(a) $𝑛$ is divisible by $3$.

(b) If $𝐵$ is another $𝑛\times 𝑛$ matrix satisfying the same equation, then $𝐴$ and $𝐵$ are similar.

(c) For $𝑛=3$, give examples of two such matrices with entries in $\mathbb{Q}$. Explain how you obtained these matrices.

Quote the relevant theorems from the theory and explain how you use them.

Let $𝐾/𝐹$ be a Galois extension with Galois group

Consider a normal subgroup $𝐻\text{⊴}𝐺$, and denote

For any two automorphisms $𝜎,𝜑\in 𝐺$, prove that their restrictions to $𝐸$ satisfy

if and only if $𝜎$ and $𝜑$ lie in the same left coset of $𝐻$ in $𝐺$.

Find a polynomial $𝑓(𝑥)\in {𝔽}_3[𝑥]$ of the form

such that ${𝔽}_{27}$ is the splitting field of $𝑓(𝑥)$ over ${𝔽}_3$. Give a complete proof that your polynomial satisfies the desired conditions.

Let $\widetilde{𝐸}/𝑘$ be a finite Galois extension of fields with Galois group $𝐺$. Let $𝐸/𝑘$ be a field extension inside $\widetilde{𝐸}$. Let ${Hom}_{𝑘}\left(𝐸,\widetilde{𝐸}\right)$ be the set of $𝑘$-algebra homomorphisms. We let $𝐺$ act on the left on the set ${Hom}_{𝑘}\left(𝐸,\widetilde{𝐸}\right)$ via $𝑔(𝜎)=𝑔\circ 𝜎$ for $𝑔\in 𝐺$ and $𝜎\in {Hom}_{𝑘}\left(𝐸,\widetilde{𝐸}\right)$.

(a) Show that there is $𝜃\in 𝐸$ such that ${\{𝜎𝜃\}}_{𝜎\in {Hom}_{𝑘}\left(𝐸,\widetilde{𝐸}\right)}$ are linearly independent over $𝑘$. (Hint: When $𝐸=\widetilde{𝐸}$, this is just the normal basis theorem, which you can use without proof.)

(b) Let $\mathrm{\Phi }\subset {Hom}_{𝑘}\left(𝐸,\widetilde{𝐸}\right)$ be a subset. Let ${𝐺}_{\mathrm{\Phi }}=\{𝑔\in 𝐺|𝑔(\mathrm{\Phi })=\mathrm{\Phi }\}\subset 𝐺$ and ${𝐸}_{\mathrm{\Phi }}={\widetilde{𝐸}}^{{𝐺}_{\mathrm{\Phi }}}$. Show that the field ${𝐸}_{\mathrm{\Phi }}$ is the subfield of $\widetilde{𝐸}$ generated by $\{{\sum }_{𝜎\in \mathrm{\Phi }}𝜎(𝑎)|𝑎\in 𝐸\}$ over $𝑘$.

(c) Show that for any such $𝑘⊊𝐸⊊\widetilde{𝐸}$ we can find $\mathrm{\Phi }$ so that neither $𝐸$ nor ${𝐸}_{\mathrm{\Phi }}$ contains the other.

Suppose $𝐿/𝐾$ is a finite Galois extension with $Gal(𝐿/𝐾)$ cyclic of degree $𝑛$ generated by $𝜎$. View $𝐿$ as a module over $𝐾[𝑥]$, where $𝑥$ acts on $𝐿$ via $𝜎$.

(a) Show that if $𝑓\in 𝐾[𝑥]$ is a non-zero polynomial, and $𝑓(𝜎)=0$ as a $𝐾$-linear endomorphism of the $𝐾$-vector space $𝐿$, then $𝑓$ has degree at least $𝑛$. (You may assume standard theorems in Galois theory if you state them clearly.) What is the minimal polynomial of $𝜎$ acting on the $𝐾$-vector space $𝐿$?

(b) Show that $𝐿=𝐾[𝑥]/({𝑥}^{𝑛}-1)$ as a $𝐾[𝑥]$-module.

(c) Prove that there is an element $𝛼\in 𝐿$ such that the set of its Galois conjugates forms a basis for $𝐿$ as a vector space over $𝐾$. (Do not just quote the Normal Basis Theorem.)

Let

be a short exact sequence of modules over a commutative ring $𝐴$, and assume ${𝑀}^{\prime }$ can be generated by ${𝑛}^{\prime }$ elements and ${𝑀}^{″}$ can be generated by ${𝑛}^{″}$ elements for integers ${𝑛}^{\prime },{𝑛}^{″}\ge 1$.

(a) For $𝑛={𝑛}^{\prime }+{𝑛}^{″}$, prove the existence and uniqueness of a linear map $𝐹:{\wedge }^{{𝑛}^{\prime }}({𝑀}^{\prime }){\otimes }_{𝐴}{\wedge }^{{𝑛}^{″}}({𝑀}^{″})\to {\wedge }^{𝑛}(𝑀)$ satisfying

for ${𝑚}_{𝑖}^{\prime }\in {𝑀}^{\prime }$ and ${𝑚}_{𝑗}^{″}\in {𝑀}^{″}$ with ${\widetilde{𝑚}}_{𝑗}^{″}\in 𝑀$ any lift of ${𝑚}_{𝑗}^{″}\in {𝑀}^{″}$. (Hint: Begin by building a multilinear map $𝜇:{𝑀}^{\prime \times {𝑛}^{\prime }}\times {𝑀}^{″\times {𝑛}^{″}}\to {\wedge }^{𝑛}(𝑀)$.)

(b) Prove $𝐹$ is surjective in general and an isomorphism when ${𝑀}^{\prime }$ and ${𝑀}^{″}$ are free with respective ranks ${𝑛}^{\prime }$ and ${𝑛}^{″}$, and give an example when $𝐹$ is not an isomorphism.

We remind you that if $𝐾$ is a field and $𝔪$ is a maximal ideal of the polynomial ring $𝐾[{𝑋}_1,\dots ,{𝑋}_{𝑟}]$, then the Nullstellensatz tells us that $𝐾[{𝑋}_1,\dots ,{𝑋}_{𝑟}]/𝔪$ is a finite field extension of $𝐾$.

Suppose that $𝐴$ is a finitely generated $\mathbb{Z}$-algebra.

(a) If $\mathbb{Z}\to 𝐴$ is injective, show that $𝐴$ cannot be a field. (Hint: if it were show that $𝐴$ would be algebraic over $\mathbb{Q}$, deduce that $𝐴$ is integral over $\mathbb{Z}[1/𝑓]$ for some positive integer $𝑓$ and derive a contradiction.)

(b) Show that if $𝔪$ is a maximal ideal of $𝐴$, then $𝐴/𝔪$ is a finite field.