##### 1

Compute each of the following Galois groups. Which of these field extensions are normal field extensions? If the extension is not normal, find a normal extension of \({\mathbb Q}\) in which the extension field is contained.

\(G({\mathbb Q}(\sqrt{30}\, ) / {\mathbb Q})\)

\(G({\mathbb Q}(\sqrt[4]{5}\, ) / {\mathbb Q})\)

\(G( {\mathbb Q}(\sqrt{2}, \sqrt{3}, \sqrt{5}\, )/ {\mathbb Q} )\)

\(G({\mathbb Q}(\sqrt{2}, \sqrt[3]{2}, i) / {\mathbb Q})\)

\(G({\mathbb Q}(\sqrt{6}, i) / {\mathbb Q})\)

##### 2

Determine the separability of each of the following polynomials.

\(x^3 + 2 x^2 - x - 2\) over \({\mathbb Q}\)

\(x^4 + 2 x^2 + 1\) over \({\mathbb Q}\)

\(x^4 + x^2 + 1\) over \({\mathbb Z}_3\)

\(x^3 +x^2 + 1\) over \({\mathbb Z}_2\)

##### 3

Give the order and describe a generator of the Galois group of \(\gf(729)\) over \(\gf(9)\).

##### 4

Determine the Galois groups of each of the following polynomials in \({\mathbb Q}[x]\); hence, determine the solvability by radicals of each of the polynomials.

\(x^5 - 12 x^2 + 2\)

\(x^5 - 4 x^4 + 2 x + 2\)

\(x^3 - 5\)

\(x^4 - x^2 - 6\)

\(x^5 + 1\)

\((x^2 - 2)(x^2 + 2)\)

\(x^8 - 1\)

\(x^8 + 1\)

\(x^4 - 3 x^2 -10\)

##### 5

Find a primitive element in the splitting field of each of the following polynomials in \({\mathbb Q}[x]\).

\(x^4 - 1\)

\(x^4 - 8 x^2 + 15\)

\(x^4 - 2 x^2 - 15\)

\(x^3 - 2\)

##### 6

Prove that the Galois group of an irreducible quadratic polynomial is isomorphic to \({\mathbb Z}_2\).

##### 7

Prove that the Galois group of an irreducible cubic polynomial is isomorphic to \(S_3\) or \({\mathbb Z}_3\).

##### 8

Let \(F \subset K \subset E\) be fields. If E is a normal extension of
\(F\), show that \(E\) must also be a normal extension of \(K\).

##### 9

Let \(G\) be the Galois group of a polynomial of degree \(n\). Prove that \(|G|\) divides \(n!\).

##### 10

Let \(F \subset E\). If \(f(x)\) is solvable over \(F\), show that \(f(x)\) is also solvable over \(E\).

##### 11

Construct a polynomial \(f(x)\) in \({\mathbb Q}[x]\) of degree 7 that is not solvable by radicals.

##### 12

Let \(p\) be prime. Prove that there exists a polynomial \(f(x) \in{\mathbb Q}[x]\) of degree \(p\) with Galois group isomorphic to \(S_p\). Conclude that for each prime \(p\) with \(p \geq 5\) there exists a polynomial of degree \(p\) that is not solvable by radicals.

##### 13

Let \(p\) be a prime and \({\mathbb Z}_p(t)\) be the field of rational functions over \({\mathbb Z}_p\). Prove that \(f(x) = x^p - t\) is an irreducible polynomial in \({\mathbb Z}_p(t)[x]\). Show that \(f(x)\) is not separable.

##### 14

Let \(E\) be an extension field of \(F\). Suppose that \(K\) and \(L\) are two intermediate fields. If there exists an element \(\sigma \in G(E/F)\) such that \(\sigma(K) = L\), then \(K\) and \(L\) are said to be *conjugate fields.* Prove that \(K\) and \(L\) are conjugate if and only if \(G(E/K)\) and \(G(E/L)\) are conjugate subgroups of \(G(E/F)\).

##### 15

Let \(\sigma \in \aut( {\mathbb R} )\). If \(a\) is a positive real number, show that \(\sigma( a) > 0\).

##### 16

Let \(K\) be the splitting field of \(x^3 + x^2 + 1 \in {\mathbb Z}_2[x]\). Prove or disprove that \(K\) is an extension by radicals.

##### 17

Let \(F\) be a field such that \({\rm char}\, F \neq 2\). Prove that the splitting field of \(f(x) = a x^2 + b x + c\) is \(F( \sqrt{\alpha}\, )\), where \(\alpha = b^2 - 4ac\).

##### 18

Prove or disprove: Two different subgroups of a Galois group will have different fixed fields.

##### 19

Let \(K\) be the splitting field of a polynomial over \(F\). If \(E\) is a field extension of \(F\) contained in \(K\) and \([E:F] = 2\), then \(E\) is the splitting field of some polynomial in \(F[x]\).

##### 20

We know that the cyclotomic polynomial
\begin{equation*}\Phi_p(x) = \frac{x^p - 1}{x - 1} = x^{p - 1} + x^{p - 2} + \cdots + x + 1\end{equation*}
is irreducible over \({\mathbb Q}\) for every prime \(p\). Let \(\omega\) be a zero of \(\Phi_p(x)\), and consider the field \({\mathbb Q}(\omega)\).

Show that \(\omega, \omega^2, \ldots, \omega^{p-1}\) are distinct zeros of \(\Phi_p(x)\), and conclude that they are all the zeros of \(\Phi_p(x)\).

Show that \(G( {\mathbb Q}( \omega ) / {\mathbb Q} )\) is abelian of order \(p - 1\).

Show that the fixed field of \(G( {\mathbb Q}( \omega ) / {\mathbb Q} )\) is \({\mathbb Q}\).

##### 21

Let \(F\) be a finite field or a field of characteristic zero. Let \(E\) be a finite normal extension of \(F\) with Galois group \(G(E/F)\). Prove that \(F \subset K \subset L \subset E\) if and only if \(\{ \identity \} \subset G(E/L) \subset G(E/K) \subset G(E/F)\).

##### 22

Let \(F\) be a field of characteristic zero and let \(f(x) \in F[x]\) be a separable polynomial of degree \(n\). If \(E\) is the splitting field of \(f(x)\), let \(\alpha_1, \ldots, \alpha_n\) be the roots of \(f(x)\) in \(E\). Let \(\Delta = \prod_{i \lt j} (\alpha_i - \alpha_j)\). We define the *discriminant* of \(f(x)\) to be \(\Delta^2\).

If \(f(x) = x^2 + b x + c\), show that \(\Delta^2 = b^2 - 4c\).

If \(f(x) = x^3 + p x + q\), show that \(\Delta^2 = - 4p^3 - 27q^2\).

Prove that \(\Delta^2\) is in \(F\).

If \(\sigma \in G(E/F)\) is a transposition of two roots of \(f(x)\), show that \(\sigma( \Delta ) = -\Delta\).

If \(\sigma \in G(E/F)\) is an even permutation of the roots of \(f(x)\), show that \(\sigma( \Delta ) = \Delta\).

Prove that \(G(E/F)\) is isomorphic to a subgroup of \(A_n\) if and only if \(\Delta \in F\).

Determine the Galois groups of \(x^3 + 2 x - 4\) and \(x^3 + x -3\).