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Section23.4Exercises

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

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

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

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

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

Hint

(a) ${\mathbb Z}_2\text{;}$ (c) ${\mathbb Z}_2 \times {\mathbb Z}_2 \times {\mathbb Z}_2\text{.}$

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 ${\mathbb Z}_2$

Hint

(a) Separable over $\mathbb Q$ since $x^3 + 2 x^2 - x - 2 = (x - 1)(x + 1)(x + 2)\text{;}$ (c) not separable over $\mathbb Z_3$ since $x^4 + x^2 + 1 = (x + 1)^2 (x + 2)^2 \text{.}$

3

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

Hint

If

\begin{equation*} [\gf(729): \gf(9)] = [\gf(729): \gf(3)] /[\gf(9): \gf(3)] = 6/2 = 3, \end{equation*}

then $G(\gf(729)/ \gf(9)) \cong {\mathbb Z}_3\text{.}$ A generator for $G(\gf(729)/ \gf(9))$ is $\sigma\text{,}$ where $\sigma_{3^6}( \alpha) = \alpha^{3^6} = \alpha^{729}$ for $\alpha \in \gf(729)\text{.}$

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. $x^5 - 12 x^2 + 2$

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

  3. $x^3 - 5$

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

  5. $x^5 + 1$

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

  7. $x^8 - 1$

  8. $x^8 + 1$

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

Hint

(a) $S_5\text{;}$ (c) $S_3\text{;}$ (g) see Example 23.10.

5

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

  1. $x^4 - 1$

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

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

  4. $x^3 - 2$

Hint

(a) ${\mathbb Q}(i)$

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 ${\mathbb Z}_3\text{.}$

Hint

Let $E$ be the splitting field of a cubic polynomial in $F[x]\text{.}$ Show that $[E:F]$ is less than or equal to 6 and is divisible by 3. Since $G(E/F)$ is a subgroup of $S_3$ whose order is divisible by 3, conclude that this group must be isomorphic to ${\mathbb Z}_3$ or $S_3\text{.}$

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 $n!\text{.}$

Hint

$G$ is a subgroup of $S_n\text{.}$

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 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.

Hint

True.

17

Let $F$ be a field such that ${\rm char}\, F \neq 2\text{.}$ 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 ${\mathbb Q}\text{.}$

Hint

  1. Clearly $\omega, \omega^2, \ldots, \omega^{p - 1}$ are distinct since $\omega \neq 1$ or 0. To show that $\omega^i$ is a zero of $\Phi_p\text{,}$ calculate $\Phi_p( \omega^i)\text{.}$

  2. The conjugates of $\omega$ are $\omega, \omega^2, \ldots, \omega^{p - 1}\text{.}$ Define a map $\phi_i: {\mathbb Q}(\omega) \rightarrow {\mathbb Q}(\omega^i)$ by

    \begin{equation*} \phi_i(a_0 + a_1 \omega + \cdots + a_{p - 2} \omega^{p - 2}) = a_0 + a_1 \omega^i + \cdots + c_{p - 2} (\omega^i)^{p - 2}, \end{equation*}

    where $a_i \in {\mathbb Q}\text{.}$ Prove that $\phi_i$ is an isomorphism of fields. Show that $\phi_2$ generates $G({\mathbb Q}(\omega)/{\mathbb Q})\text{.}$

  3. Show that $\{ \omega, \omega^2, \ldots, \omega^{p - 1} \}$ is a basis for ${\mathbb Q}( \omega )$ over ${\mathbb Q}\text{,}$ and consider which linear combinations of $\omega, \omega^2, \ldots, \omega^{p - 1}$ are left fixed by all elements of $G( {\mathbb Q}( \omega ) / {\mathbb Q})\text{.}$

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 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 $x^3 + x -3\text{.}$