querent

Let $𝑝$ be a prime number. Show that a free $\mathbb{Z}$-module of rank $2$ has $𝑝+1$ submodules of index $𝑝$.

Let $𝑅$ be a commutative Noetherian ring in which each ideal $𝐼$ is principal and satisfies ${𝐼}^2=𝐼$. Show that $𝑅$ is isomorphic to a finite product of fields.

Let $𝛼=1+\sqrt[3]{2}+\sqrt[3]{4}$.

(a) Find the degree of $𝛼$ over $\mathbb{Q}$. Justify your answer.

(b) Find a normal closure of $\frac{\mathbb{Q}(𝛼)}{\mathbb{Q}}$. Justify your answer.

Let $𝑞$ be a power of a prime integer, $𝑛\in \mathbb{N}$. Let $𝑘$ be the least positive integer such that ${𝑞}^{𝑘}\cong 1(mod𝑛)$. Prove that the finite field ${𝔽}_{{𝑞}^{𝑘}}$ is a splitting field of the polynomial ${𝑋}^{𝑛}-1$ over ${𝔽}_{𝑞}$.

Find a subfield $𝐹$ in the field of rational functions $ℂ(𝑋)$ such that $\frac{ℂ(𝑋)}{𝐹}$ is a Galois extension with the Galois group isomorphic to the symmetric group ${𝑆}_3$. (Hint: Consider automorphisms of $ℂ(𝑋)$ given by $𝑋⟼\frac{𝑎𝑋+𝑏}{𝑐𝑋+𝑑}$, $𝑎,𝑏,𝑐,𝑑\in \mathbb{Z}$.)

Let $𝐴$ be a linear operator on a vector space of dimension $𝑛$ such that ${𝐴}^{𝑚}$ is the zero operator for some $𝑚$.

(a) Prove that all eigenvalues of $𝐴$ are equal to zero.

(b) Prove that ${𝐴}^{𝑛}=0$.

Describe, up to similarity, all $4\times 4$ matrices $𝐴$ over $\mathbb{Q}$ such that ${𝐴}^5=-{𝐴}^3$ but ${𝐴}^3\ne 0$. Justify your answer.

Let $End(𝑉)$ be the ring of all linear operators on a finite dimensional vector space $𝑉$ (with respect to the addition and composition of operators). For an operator $𝐴\in End(𝑉)$ let ${𝐿}_{𝐴}$ be the subspace in $End(𝑉)$ generated by the powers ${𝐴}^{𝑖}$, $𝑖\ge 0$.

(a) Show that ${𝐿}_{𝐴}$ is a subring in $End(𝑉)$.

(b) Prove that if ${𝐿}_{𝐴}$ is a field then the characteristic polynomial of $𝐴$ is a power of an irreducible polynomial.

(a) Explicitly exhibit an element $𝜎$ of $𝐺=GL(2,\mathbb{Z}/7\mathbb{Z})$ of order 8.

(b) Describe, with proof, the structure of the 2-Sylow subgroup of $𝐺$. Hint: think about the multiplicative group of the field of size 49, and the action of the nontrivial automorphism of this field.

Find a root of unity $𝜁$ so that $\mathbb{Q}(𝜁)$ contains a subfield $𝐾$ which is Galois over $\mathbb{Q}$ with Galois group $\mathbb{Z}/3\mathbb{Z}$. Compute the minimal polynomial over $\mathbb{Q}$ of an element that generates $𝐾$ over $\mathbb{Q}$.