An Introduction to Galois Theory

Let $E$ be an extension field of $\mathbb {Q}$. Show that any automorphism of $E$ acts as the identity on $\mathbb {Q}$. (This exercise is referred to in this chapter.)

Determine the group of field automorphisms of $\mathrm{GF}(4)$.

Let $E$ be an extension field of the field $F$. Show that the automorphism group of $E$ fixing $F$ is indeed a group. (This exercise is referred to in this chapter.)

Given that the automorphism group of $\mathbb {Q}(\sqrt{2}, \sqrt{5}, \sqrt{7})$ is isomorphic to $Z_2 \oplus Z_2 \oplus Z_2$, determine the number of subfields of $\mathbb {Q}(\sqrt{2}, \sqrt{5}, \sqrt{7})$ that have degree $4$ over $\mathbb {Q}$.

Let $E$ be an extension field of a field $F$ and let $H$ be a subgroup of $\mathrm{Gal}(E/F)$. Show that the fixed field of $H$ is indeed a field. (This exercise is referred to in this chapter.)

Let $E$ be the splitting field of $x^4 + 1$ over $\mathbb {Q}$. Find $\mathrm{Gal}(E/\mathbb {Q})$. Find all subfields of $E$. Find the automorphisms of $E$ that have fixed fields $\mathbb {Q}(\sqrt{2})$, $\mathbb {Q}(\sqrt{-2})$, and $\mathbb {Q}(i)$. Is there an automorphism of $E$ whose fixed field is $\mathbb {Q}$?

Let $f(x) \in F[x]$ and let the zeros of $f(x)$ be $a_1, a_2, \ldots , a_n$. If $K = F(a_1, a_2, \ldots , a_n)$, show that $\mathrm{Gal}(K/F)$ is isomorphic to a group of permutations of the $a_i$'s.

Show that the Galois group of a polynomial of degree $n$ has order dividing $n!$

Referring to Example 6, show that the automorphism $\phi$ has order $6$. Show that $\omega + \omega^{-1}$ is fixed by $\phi^3$ and $\omega^3 + \omega^5 + \omega^6$ is fixed by $\phi^2$. (This exercise is referred to in this chapter.)

Let $E = \mathbb {Q}(\sqrt{2}, \sqrt{5})$. What is the order of the group $\mathrm{Gal}(E/\mathbb {Q})$? What is the order of $\mathrm{Gal}(\mathbb {Q}(\sqrt{10})/\mathbb {Q})$?

Suppose that $F$ is a field of characteristic $0$ and $E$ is the splitting field for some polynomial over $F$. If $\mathrm{Gal}(E/F)$ is isomorphic to $Z_{20} \oplus Z_2$, determine the number of subfields $L$ of $E$ there are such that $L$ contains $F$ and

Determine the Galois group of $x^2 - 10x + 21$ over $\mathbb {Q}$. (See Exercise 7 for the definition).

Determine the Galois group of $x^2 + 9$ over $\mathbb {R}$. (See Exercise 7 for the definition).

Suppose that $F$ is a field of characteristic $0$ and $E$ is the splitting field for some polynomial over $F$. If $\mathrm{Gal}(E/F)$ is isomorphic to $D_6$, prove that there are exactly three fields $L$ such that $E \supseteq L \supseteq F$ and $[E : L] = 6$.

Suppose that $E$ is the splitting field for some polynomial over $\mathrm{GF}(p)$. If $\mathrm{Gal}(E/\mathrm{GF}(p)) = p^6$, how many fields are there strictly between $E$ and $\mathrm{GF}(p)$?

Let $p$ be a prime. Suppose that $|\mathrm{Gal}(E/F)| = p^2$. Draw all possible subfield lattices for fields between $E$ and $F$.

Suppose that $F$ is a field of characteristic $0$ and $E$ is the splitting field for some polynomial over $F$. If $\mathrm{Gal}(E/F)$ is isomorphic to $A_4$, show that there is no subfield $K$ of $E$ such that $[K : F] = 2$.

Determine the Galois group of $x^3 = 1$ over $\mathbb {Q}$ and $x^3 - 2$ over $\mathbb {Q}$. (See Exercise 7 for the definition.)

Suppose that $K$ is the splitting field of some polynomial over a field $F$ of characteristic $0$. If $[K : F] = p^2q$, where $p$ and $q$ are distinct primes, show that $K$ has subfields $L_1, L_2$, and $L_3$ such that $[K : L_1] = p$, $[K : L_2] = p^2$, and $[K : L_3] = q$.

Suppose that $E$ is the splitting field of some polynomial over a field $F$ of characteristic $0$. If $\mathrm{Gal}(E/F)$ is isomorphic to $D_5$, draw the subfield lattice for the fields between $E$ and $F$.

Suppose that $F \subset K \subset E$ are fields and $E$ is the splitting field of some polynomial in $F[x]$. Show, by means of an example, that $K$ need not be the splitting field of some polynomial in $F[x]$.

Suppose that $E$ is the splitting field of some polynomial over a field $F$ of characteristic $0$. If $[E : F]$ is finite, show that there is only a finite number of fields between $E$ and $F$.

Suppose that $E$ is the splitting field of some polynomial over a field $F$ of characteristic $0$. If $\mathrm{Gal}(E/F)$ is an Abelian group of order $10$, draw the subfield lattice for the fields between $E$ and $F$.

Let $\omega$ be a nonreal complex number such that $\omega^5 = 1$. If $\phi$ is the automorphism of $\mathbb {Q}(\omega )$ that carries $\omega$ to $\omega^4$, find the fixed field of $\langle \phi \rangle$.

Determine the isomorphism class of the group $\mathrm{Gal}(\mathrm{GF}(64)/ \mathrm{GF}(2))$.

Determine the isomorphism class of the group $\mathrm{Gal}(\mathrm{GF}(729)/ \mathrm{GF}(9))$.

Exercises 27, 28, and 29 are referred to in this chapter.

Show that $S_5$ is not solvable.

Show that the dihedral groups are solvable.

Show that a group of order $p^n$, where $p$ is prime, is solvable.

Show that $S_n$ is solvable when $n \le 4$.

Complete the proof of Theorem 30.3 by showing that the given series of groups satisfies the definition for solvability.

Show that a subgroup of a solvable group is solvable.

Let $N$ be a normal subgroup of $G$ and let $K/N$ be a normal subgroup of $G/N$. Prove that $K$ is a normal subgroup of $G$. (This exercise is referred to in this chapter.)

Show that any automorphism of $\mathrm{GF}(p^n)$ acts as the identity on $\mathrm{GF}(p)$.

If $G$ is a finite solvable group, show that there exist subgroups of $G$

such that $H_{i+1}/H_i$ has prime order.

Show that the polynomial $x^5 - 6x + 3$ over $\mathbb {Q}$ is not solvable by radicals.