Galois Theory

Consider the field extension $\mathbb{C}/\mathbb{R}$.

1. Find the order of $\mathrm{Gal}(\mathbb{C}/\mathbb{R})$.

2. Find the element(s) of $\mathrm{Gal}(\mathbb{C}/\mathbb{R})$.

Prove 1.16.

Prove 1.20.

Let $E$ be a field, $G$ be a subgroup of $\mathrm{Aut}(E)$, and $F=\mathrm{Fix}_{E}(G)$. Show that $G\leq\mathrm{Gal}(E/F)$.

Prove that the group action defined in 1.23 is indeed a group action.

Prove that the degree of $\alpha_{1}$ over $F$ in 1.23 divides $|G|$.

Prove 1.28.

Prove that all abelian groups are solvable.

Prove 1.56.

Let $E$ be the splitting field of a polynomial of degree $n$ over the field $F$. Prove that $|\mathrm{Gal}(E/F)|$ divides $n!$.

Let $f(x)=x^{2}+3$ and $F$ be the splitting field of $f(x)$ over $\mathbb{R}$. Determine $F$ and $\mathrm{Gal}(F/\mathbb{R})$.

Let $F$ be a field, $f(x)$ an irreducible quadratic polynomial over $F$, and $E$ be its splitting field over $F$. Prove that $\mathrm{Gal}(E/F)\cong\mathrm{C}_{2}$.

Let $\sigma\in\mathrm{Aut}(\mathbb{R})$. Prove that $\sigma(x)>0$ for all $x>0$.

Suppose $E$ is the splitting field of some polynomial over $\mathrm{GF}(p)$, where $p$ is a prime. If $|\mathrm{Gal}(E/\mathrm{GF}(p))|=q^{6}$ where $q$ is also a prime, how many fields are strictly between $E$ and $\mathrm{GF}(p)$? That is, how many fields $F$ are there such that $\mathrm{GF}(p)\subset F\subset E$?

Consider the polynomial $f(x)=x^{3}-2$. Let $E$ be the splitting field of $f(x)$ over $\mathbb{Q}$.

1. Find a field $F$ such that $F$ is a simple extension of $\mathbb{Q}$, its primitive element is a real number, and $\mathbb{Q}\subset F\subset E$.

2. Show that $F$ is not a splitting field of any polynomial in $\mathbb{Q}[x]$.

Prove that $D_{n}$ is solvable for all positive integers $n$.

Prove that $\mathrm{S}_{n}$ is not solvable for all $n\geq 5$.

Let $\theta\in\mathbb{R}$. De Moivre’s formula states that

for all non-negative integers $n$.

1. Prove De Moivre’s formula.(Hint: you will need an exercise given elsewhere in the book and an exercise given elsewhere in the book.)

2. Deduce a primitive $n$th root of unity in $\mathbb{C}$, proving your answer.

Let $F$ be a field and $f(x)\in F[x]$ be a separable polynomial of degree $n$. Suppose $E$ is the splitting field of $f(x)$ over $F$. Let $\alpha_{1},\alpha_{2},\dots,\alpha_{n}$ be the zeroes of $f(x)$ in $E$. Define

For example, for two distinct zeroes, we have $D=\alpha_{1}-\alpha_{2}$. For three distinct zeroes, we have $D=(\alpha_{1}-\alpha_{2})(\alpha_{1}-\alpha_{3})(\alpha_{2}-\alpha_{3})$. The discriminant of $f(x)$ is defined to be $\Delta=D^{2}$.

1. Verify that the discriminant of $x^{4}-6x^{3}+11x^{2}-6x$ is 144 for the case where $F=\mathbb{Q}$.

2. Prove that $\Delta\in F$.(Hint: consider what an automorphism from $\mathrm{Gal}(E/F)$…