3.8 Exercises ‣ Chapter 3 The Fundamental Theorem of Galois Theory

3-1

Let $F$ be a field of characteristic $0$ . Show that $F(X^{2})\cap F(X^{2}-X)=F$ (intersection inside $F(X)$ ). [Hint: Find automorphisms $\sigma$ and $\tau$ of $F(X)$ , each of order $2$ , fixing $F(X^{2})$ and $F(X^{2}-X)$ respectively, and show that $\sigma\tau$ has infinite order.]

3-2

⁴⁴ 4 This problem shows that every quadratic extension of

{\mathbb{Q}}

is contained in a cyclotomic extension of

{\mathbb{Q}}

. The Kronecker-Weber theorem says that every abelian extension of

{\mathbb{Q}}

is contained in a cyclotomic extension.

Let $p$ be an odd prime, and let $\zeta$ be a primitive $p$ th root of $1$ in $\mathbb{C}$ . Let $E={\mathbb{Q}}[\zeta]$ , and let $G=\operatorname{Gal}(E/{\mathbb{Q}})$ ; thus $G=(\mathbb{Z}/(p))^{\times}$ . Let $H$ be the subgroup of index $2$ in $G$ . Put $\alpha=\sum_{i\in H}\zeta^{i}$ and $\beta=\sum_{i\in G\setminus H}\zeta^{i}$ . Show:

(a)

$\alpha$ and $\beta$ are fixed by $H$ ;
(b)

if $\sigma\in G\setminus H$ , then $\sigma\alpha=\beta$ , $\sigma\beta=\alpha$ .

Thus $\alpha$ and $\beta$ are roots of the polynomial $X^{2}+X+\alpha\beta\in{\mathbb{Q}}[X]$ . Compute⁵⁵ 5 Schoof suggests computing $\alpha-\beta$ instead. $\alpha\beta$ and show that the fixed field of $H$ is ${\mathbb{Q}}[\sqrt{p}]$ when $p\equiv 1\,\mod 4$ and ${\mathbb{Q}}[\sqrt{-p}]$ when $p\equiv 3\mod 4$ .

3-3

Let $M={\mathbb{Q}}[\sqrt{2},\sqrt{3}]$ and $E=M[\sqrt{(\sqrt{2}+2)(\sqrt{3}+3)}]$ (subfields of $\mathbb{R}$ ).

(a)

Show that $M$ is Galois over ${\mathbb{Q}}$ with Galois group the $4$ -group $C_{2}\times C_{2}$ .
(b)

Show that $E$ is Galois over ${\mathbb{Q}}$ with Galois group the quaternion group.

3-4

Let $E$ be a Galois extension of $F$ with Galois group $G$ , and let $L$ be the fixed field of a subgroup $H$ of $G$ . Show that the automomorphism group of $L/F$ is $N/H$ where $N$ is the normalizer of $H$ in $G$ .

3-5

Let $E$ be a finite extension of $F$ . Show that the order of $\operatorname{Aut}(E/F)$ divides the degree $[E\colon F].$