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Let $𝐸$ be the splitting field of

over $\mathbb{Q}$. Let $𝜁$ be a zero of $𝑓(𝑥)$, i.e. a primitive seventh root of 1.

(a) Show that $𝑓(𝑥)$ is irreducible over $\mathbb{Q}$. (Hint: consider $𝑓(𝑦+1)$ and use Eisenstein's criterion.)

(b) Show that the Galois group of $𝐸/\mathbb{Q}$ is cyclic, and find an explicit generator.

(c) Let $𝛽=𝜁+{𝜁}^2+{𝜁}^4$. Show that the intermediate field $\mathbb{Q}(𝛽)$ is actually $\mathbb{Q}\left(\sqrt{-7}\right)$. (Hint: first show that $[\mathbb{Q}(𝛽):\mathbb{Q}]=2$ by finding a linear dependence over $\mathbb{Q}$ among $\{1,𝛽,{𝛽}^2\}$.)

(d) Let ${𝛾}_{𝑞}=𝜁+{𝜁}^{𝑞}$. Find (with proof) a $𝑞$ such that $\mathbb{Q}({𝛾}_{𝑞})$ is a degree 3 extension of $\mathbb{Q}$. (Hint: use (b).) Is this extension Galois?

Let $𝑉$ be a vector space of finite dimension $𝑑\ge 1$ over a field $𝑘$ of arbitrary characteristic. Let ${𝑉}^{\ast }$ denote the dual space.

(a) For any $𝑛\ge 1$, prove that there is a unique bilinear pairing ${𝑉}^{\otimes 𝑛}\times {({𝑉}^{\ast })}^{\otimes 𝑛}\to 𝑘$ satisfying

and by using bases show that it is a perfect pairing (i.e., identifies ${({𝑉}^{\ast })}^{\otimes 𝑛}$ with ${({𝑉}^{\otimes 𝑛})}^{\ast }$).

(b) For any $1\le 𝑛\le 𝑑$, do similarly with ${\wedge }^{𝑛}(𝑉)$ and ${\wedge }^{𝑛}({𝑉}^{\ast })$ using the requirement

Let $𝐺$ be a nontrivial finite group and $𝑝$ be the smallest prime dividing the order of $𝐺$. Let $𝐻$ be a subgroup of index $𝑝$. Show that $𝐻$ is normal. (Hint: If $𝐻$ isn't normal, consider the action of $𝐺$ on the conjugates of $𝐻$.)

Let $𝐾/𝑘$ be a finite extension of fields with $𝛼\in 𝐾$ as a primitive element over $𝑘$. Let $𝑓\in 𝑘[𝑥]$ be the minimal polynomial of $𝛼$ over $𝑘$.

(a) Explain why $𝐾\cong 𝑘[𝑥]/(𝑓)$ as $𝑘$-algebras, and use this to relate the local factor rings of $𝐾{\otimes }_{𝑘}𝐹$ to the irreducible factors of $𝑓$ in $𝐹[𝑥]$, with $𝐹/𝑘$ a field extension.

(b) Assume $𝐾/𝑘$ is Galois with Galois group $𝐺$. Prove that the natural map $𝐾{\otimes }_{𝑘}𝐾\to {\prod }_{𝑔\in 𝐺}𝐾$ defined by $𝑎\otimes 𝑏\mapsto (𝑔(𝑎)𝑏)$ is an isomorphism.

Let $𝐺$ be a finite group and $𝜋:𝐺\to GL(𝑉)$ a finite-dimensional complex representation. Let $𝜒$ be the character of $𝜋$. Show that the characters of the representations on $𝑉\otimes 𝑉$, ${Sym}^2(𝑉)$ and ${\wedge }^2(𝑉)$ are ${𝜒(𝑔)}^2$, $({𝜒(𝑔)}^2+𝜒({𝑔}^2))/2$ and $({𝜒(𝑔)}^2-𝜒({𝑔}^2))/2$. (Hint: Express ${𝜒(𝑔)}^2$, $({𝜒(𝑔)}^2+𝜒({𝑔}^2))/2$ and $({𝜒(𝑔)}^2-𝜒({𝑔}^2))/2$ in terms of the eigenvalues of $𝜋(𝑔)$.)

Let $𝐺$ be a finite abelian group, $𝜔:𝐺\times 𝐺\to ℝ/\mathbb{Z}$ a bilinear mapping so that (i) $𝜔(𝑔,𝑔)=0$ for all $𝑔$ in $𝐺$; (ii) $𝜔(𝑥,𝑔)=0$ for all $𝑔$ if and only if $𝑥$ is the identity element. Prove that the order of $𝐺$ is a square. Give an example of $𝐺$ of square order for which no such $𝜔$ exists.

Hint: Consider a subgroup $𝐴$ of $𝐺$ which is maximal for the property that $𝜔(𝑥,𝑦)=0$ for $𝑥,𝑦$ in $𝐴$. You may use the following fact without proof: any finite abelian group $𝑋$ admits $|𝑋|$ distinct homomorphisms to $ℝ/\mathbb{Z}$.

Let $𝐺$ be a subgroup of a finite $𝑝$-group $𝐻$ ($𝑝$ a prime) such that the natural homomorphism $𝐺\to 𝐻/[𝐻,𝐻]$ is surjective. Prove that $𝐺=𝐻$ by induction on $|𝐻|$ as follows:

(i) Suppose $𝑁$ is any nontrivial normal subgroup of $𝐻$; show (using the inductive assumption) that $𝐺\cdot 𝑁=𝐻$.

(ii) Let $𝑍$ be the center of $𝐻$. Using (i) we have $𝐺\cdot 𝑍=𝐻$; explain why $𝐺\cap 𝑍$ cannot be trivial. Now set $𝑁=𝐺\cap 𝑍$ in (i).

(i) Let $𝐺$ be a finite group of order $𝑛$, and let $𝜌:𝐺\to Sym(𝐺)$ be the homomorphism that arises from $𝐺$ acting on itself by left translation. Let $𝑔\in 𝐺$ have order $𝑚$. Prove that the sign of $𝜌(𝑔)$ is ${(-1)}^{𝑛+𝑛/𝑚}$.

(ii) Let $𝐺$ be a finite group of order $2𝑘$, with $𝑘$ odd. Prove that $𝐺$ is a semidirect product $𝑁⋊(\mathbb{Z}/2\mathbb{Z})$, where $𝑁$ has order $𝑘$. Hint: Using part (i), construct a nontrivial homomorphism $𝐺\to \mathbb{Z}/2\mathbb{Z}$.

Let $𝐺$ be a finite group and $𝐾\subset 𝐿$ an extension of fields of any characteristic. For a $𝐾$-vector space $𝑊$, let ${𝑊}_{𝐿}$ denote $𝐿{\otimes }_{𝐾}𝑊$. Let $𝑉,{𝑉}^{\prime }$ be $𝑛$-dimensional $𝐾$-linear representations of $𝐺$ (with $𝑛\ge 1$).

(i) Prove that there is a nonzero $𝐾[𝐺]$-linear map ${𝑉}^{\prime }\to 𝑉$ if there is a nonzero $𝐿[𝐺]$-linear map ${𝑉}_{𝐿}^{\prime }\to {𝑉}_{𝐿}$.

(ii) Prove (for any $𝑁\ge 1$) that a nonzero polynomial over $𝐾$ in $𝑁$ variables cannot vanish on ${𝐾}^{𝑁}$ if $𝐾$ is infinite, and deduce that if $𝐾$ is infinite then

Let $𝐺$ be a group, and $𝑘$ a field.

(i) Define the group algebra $𝑘[𝐺]$, and briefly explain why a $𝑘$-linear representation of $𝐺$ is "the same" as a left $𝑘[𝐺]$-module.

(ii) Explain how to make $𝐴{\otimes }_{𝑘}𝐵$ naturally into an associative $𝑘$-algebra for any two associative $𝑘$-algebras $𝐴$ and $𝐵$, and construct a natural $𝑘$-algebra isomorphism $𝑘[𝐺]{\otimes }_{𝑘}𝑘[𝐻]\cong 𝑘[𝐺\times 𝐻]$ for any two groups $𝐺$ and $𝐻$.

(iii) For an associative $𝑘$-algebra $𝐴$, define its opposite algebra ${𝐴}^{\text{opp}}$ to have underlying $𝑘$-vector space $𝐴$ but the multiplication law $𝑎\times {𝑎}^{\prime }≔{𝑎}^{\prime }𝑎$ ("flipped around"). Explain briefly why this is is an associative $𝑘$-algebra, and for $𝐴=𝑘[𝐺]$ prove that ${𝐴}^{\text{opp}}\cong 𝐴$. (Hint: use inversion in $𝐺$).