## Exercises23.5Exercises

### 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.

1. $$\displaystyle G({\mathbb Q}(\sqrt{30}\, ) / {\mathbb Q})$$

2. $$\displaystyle G({\mathbb Q}(\sqrt{5}\, ) / {\mathbb Q})$$

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

4. $$\displaystyle G({\mathbb Q}(\sqrt{2}, \sqrt{2}, i) / {\mathbb Q})$$

5. $$\displaystyle G({\mathbb Q}(\sqrt{6}, i) / {\mathbb Q})$$

### 2.

Determine the separability of each of the following polynomials.

1. $$x^3 + 2 x^2 - x - 2$$ over $${\mathbb Q}$$

2. $$x^4 + 2 x^2 + 1$$ over $${\mathbb Q}$$

3. $$x^4 + x^2 + 1$$ over $${\mathbb Z}_3$$

4. $$x^3 +x^2 + 1$$ over $${\mathbb Z}_2$$

### 3.

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

### 4.

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

1. $$\displaystyle x^5 - 12 x^2 + 2$$

2. $$\displaystyle x^5 - 4 x^4 + 2 x + 2$$

3. $$\displaystyle x^3 - 5$$

4. $$\displaystyle x^4 - x^2 - 6$$

5. $$\displaystyle x^5 + 1$$

6. $$\displaystyle (x^2 - 2)(x^2 + 2)$$

7. $$\displaystyle x^8 - 1$$

8. $$\displaystyle x^8 + 1$$

9. $$\displaystyle x^4 - 3 x^2 -10$$

### 5.

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

1. $$\displaystyle x^4 - 1$$

2. $$\displaystyle x^4 - 8 x^2 + 15$$

3. $$\displaystyle x^4 - 2 x^2 - 15$$

4. $$\displaystyle x^3 - 2$$

### 6.

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

### 7.

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

### 8.

Let $$F \subset K \subset E$$ be fields. If $$E$$ is a normal extension of $$F\text{,}$$ show that $$E$$ must also be a normal extension of $$K\text{.}$$

### 9.

Let $$G$$ be the Galois group of a polynomial of degree $$n\text{.}$$ Prove that $$|G|$$ divides $$n!\text{.}$$

### 10.

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

### 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 $$S_p\text{.}$$ 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 $${\mathbb Z}_p\text{.}$$ Prove that $$f(x) = x^p - t$$ is an irreducible polynomial in $${\mathbb Z}_p(t)[x]\text{.}$$ Show that $$f(x)$$ is not separable.

### 14.

Let $$E$$ be an extension field of $$F\text{.}$$ Suppose that $$K$$ and $$L$$ are two intermediate fields. If there exists an element $$\sigma \in G(E/F)$$ such that $$\sigma(K) = L\text{,}$$ 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 $$G(E/F)\text{.}$$

### 15.

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

### 16.

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

### 17.

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

### 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 $$F\text{.}$$ If $$E$$ is a field extension of $$F$$ contained in $$K$$ and $$[E:F] = 2\text{,}$$ then $$E$$ is the splitting field of some polynomial in $$F[x]\text{.}$$

### 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 $$p\text{.}$$ Let $$\omega$$ be a zero of $$\Phi_p(x)\text{,}$$ and consider the field $${\mathbb Q}(\omega)\text{.}$$

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

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

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

### 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 $$G(E/F)\text{.}$$ Prove that $$F \subset K \subset L \subset E$$ if and only if $$\{ \identity \} \subset G(E/L) \subset G(E/K) \subset G(E/F)\text{.}$$

### 22.

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

1. If $$f(x) = x^2 + b x + c\text{,}$$ show that $$\Delta^2 = b^2 - 4c\text{.}$$

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

3. Prove that $$\Delta^2$$ is in $$F\text{.}$$

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

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

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

7. Determine the Galois groups of $$x^3 + 2 x - 4$$ and $$x^3 + x -3\text{.}$$