querent

Suppose $1\to 𝑁\stackrel{𝑖}{\to }𝐺\stackrel{𝑗}{\to }𝐾\to 1$ is an exact sequence of groups, with $𝐺$ finite. Let $𝑃\subset 𝐺$ be a $𝑝$-Sylow subgroup.

(a) Show that $𝑗(𝑃)$ is a $𝑝$-Sylow subgroup of $𝐾$.

(b) If ${𝑃}_1,{𝑃}_2$ are two $𝑝$-Sylow subgroups of $𝐺$ with $𝑗({𝑃}_1)=𝑗({𝑃}_2)$, show that there exists $𝑛\in 𝑁$ with $𝑖(𝑛){𝑃}_2{𝑖(𝑛)}^{-1}={𝑃}_1$. (Hint: apply a Sylow theorem to a subgroup of $𝐺$.)

(a) Let $𝜁$ denote a complex primitive ${25}^{\text{th}}$ root of unity. Show that ${𝑥}^5-5$ has no roots in $\mathbb{Q}[𝜁]$.

(b) If ${𝛼}^5=5$, show that $𝛼$ is not a $5^{\text{th}}$ power in $\mathbb{Q}[𝜁,𝛼]$.

Let $𝐴$ be a commutative ring, and $𝑀$ an $𝐴$-module.

(a) Define what it means to say that $𝑀$ is $𝐴$-flat, and prove that $\mathbb{Q}$ is a flat $\mathbb{Z}$-module that is not projective.

(b) Prove that $𝑀$ is flat if and only if ${Tor}_{𝐴}^1(𝑀,𝑁)=0$ for all $𝐴$-modules $𝑁$.

(c) Prove that if $0\to {𝑀}^{\prime }\to 𝑀\to {𝑀}^{″}\to 0$ is a short exact sequence of $𝐴$-modules and ${𝑀}^{\prime }$ and ${𝑀}^{″}$ are $𝐴$-flat then so is $𝑀$.

(a) Let $𝑞={𝑝}^{𝑛}$, $𝑝$ prime, and let ${𝔽}_{𝑞}$ denote a finite field of $𝑞$ elements. How many monic irreducible polynomials of degree 2 are there over ${𝔽}_{𝑞}$? How many monic irreducible polynomials of degree 3 are there over ${𝔽}_{𝑞}$? (Hint: Think about elements of ${𝔽}_{{𝑞}^2}$ and ${𝔽}_{{𝑞}^3}$.)

(b) Determine the number of conjugacy classes in the group ${GL}_3({𝔽}_{𝑞})$. (Hint: Use canonical forms of modules over a principal ideal domain. One canonical form would use part (a), but you can also solve part (b) without using part (a).)

Let be an algebraic closure of a field $𝐹$. Consider a field $𝐾$ with and a homomorphism with ${𝜙}_0{|}_{𝐹}=Id$. Prove that ${𝜙}_0$ extends to an automorphism . (Hint: first produce a homomorphism , and then show that $𝜙$ is surjective.)

(a) Let $𝐾$ be an algebraically closed field. Suppose $𝑆\subset {𝐾}^{𝑛}$ is the set of common zeros of a family of polynomials $\{{𝑓}_{𝑖}\}\subset 𝐾[{𝑥}_1,\dots ,{𝑥}_{𝑛}]$, and assume $𝑆$ is non-empty. Suppose

is a rational function such that the polynomial $𝑑$ is non-zero at all points of $𝑆$. Thus $𝑟$ defines a $𝐾$-valued function on $𝑆$. Prove that there is a polynomial $ℎ\in 𝐾[{𝑥}_1,\dots ,{𝑥}_{𝑛}]$ so that $ℎ\left(\overrightarrow{𝑥}\right)=𝑟\left(\overrightarrow{𝑥}\right)$ for all $\overrightarrow{𝑥}\in 𝑆$. (Hint: Consider the ideal generated by the ${𝑓}_{𝑖}$ and $𝑑$.)

(b) Give a counterexample to part (a) if $𝐾$ is not algebraically closed, by taking $𝐾=\mathbb{Q}$, $𝑛=2$, ${𝑓}_1={𝑥}^2+{𝑦}^2-1$, and $𝑟=\frac{1}{𝑦-𝑥}$, and showing that there is no $ℎ\in \mathbb{Q}[𝑥,𝑦]$ with $ℎ(𝑥,𝑦)=𝑟(𝑥,𝑦)$ on $𝑆=\{(𝑥,𝑦)\in {\mathbb{Q}}^2:{𝑥}^2+{𝑦}^2=1\}$. (Hint: You may use without proof the fact that any polynomial $𝑔\in \mathbb{Q}[𝑥,𝑦]$ that vanishes on $𝑆$ must be a multiple of ${𝑥}^2+{𝑦}^2-1$.)

Let $𝜋:𝐺\to GL(𝑉)$ be a finite-dimensional complex representation of a finite group $𝐺$. On the respective spaces $Bil(𝑉)$ and $Hom(𝑉,{𝑉}^{\ast })$ of bilinear forms (on $𝑉$) and linear maps, define left $𝐺$-actions

(a) Prove that the natural linear map $Bil(𝑉)\to Hom(𝑉,{𝑉}^{\ast })$ defined by $𝐵\mapsto (𝑣\mapsto 𝐵(𝑣,\cdot ))$ is an isomorphism as well as $𝐺$-equivariant.

(b) Prove that ${Hom}_{ℂ[𝐺]}(𝑉,{𝑉}^{\ast })\ne 0$ if and only if there exists a nonzero bilinear form $𝐵:𝑉\times 𝑉\to ℂ$ satisfying $𝐵(𝑔(𝑣),𝑔({𝑣}^{\prime }))=𝐵(𝑣,{𝑣}^{\prime })$ for all $𝑔\in 𝐺$ and $𝑣,{𝑣}^{\prime }\in 𝑉$, and deduce that if $𝑉$ is irreducible then such a nonzero $𝐵$ exists if and only if the character of $𝜋$ is $ℝ$-valued.

Find, with proof, all algebraic integers in the field $\mathbb{Q}\left[\sqrt{6}\right]$. For which of the integer primes $𝑝=2,3,5,7,11$ is there exactly one prime ideal in the ring of integers lying over the prime ideal $(𝑝)\subset \mathbb{Z}$?

Let $𝑅$ be a finite-dimensional algebra over a field $𝑘$.

(a) Prove that if $𝑅$ is a commutative integral domain, then $𝑅$ is a field.

(b) Suppose $𝑅$ is not commutative. Prove that $𝑟𝑠=1$ implies $𝑠𝑟=1$.

(a) Let $𝐺$ be a group and $𝐻$ a subgroup of finite index $𝑛>0$. Prove that $𝐺$ contains a normal subgroup of index at most $𝑛!$. (Hint: think about homomorphisms from $𝐺$ to ${𝑆}_{𝑛}$.)

(b) Let $𝐺$ be a group which is generated by two elements. Prove that $𝐺$ has at most 17 subgroups of index 3. (Hint: think about homomorphisms from $𝐺$ to ${𝑆}_3$.)