{ "cells": [ {"cell_type": "code", "execution_count":null, "metadata":{}, "source":["%%html\n\n\n\n\n\n\n"], "outputs":[]}, {"cell_type":"markdown", "metadata":{}, "source":["**Important:** to view this notebook properly you will need to execute the cell above, which assumes you have an Internet connection. It should already be selected, or place your cursor anywhere above to select. Then press the \"Run\" button in the menu bar above (the right-pointing arrowhead), or press Shift-Enter on your keyboard."]}, {"cell_type":"markdown", "metadata":{}, "source":["$\\newcommand{\\identity}{\\mathrm{id}}\n\\newcommand{\\notdivide}{\\nmid}\n\\newcommand{\\notsubset}{\\not\\subset}\n\\newcommand{\\lcm}{\\operatorname{lcm}}\n\\newcommand{\\gf}{\\operatorname{GF}}\n\\newcommand{\\inn}{\\operatorname{Inn}}\n\\newcommand{\\aut}{\\operatorname{Aut}}\n\\newcommand{\\Hom}{\\operatorname{Hom}}\n\\newcommand{\\cis}{\\operatorname{cis}}\n\\newcommand{\\chr}{\\operatorname{char}}\n\\newcommand{\\Null}{\\operatorname{Null}}\n\\newcommand{\\lt}{<}\n\\newcommand{\\gt}{>}\n\\newcommand{\\amp}{&}\n$"]}, {"cell_type":"markdown", "metadata":{}, "source":["

Section23.4Exercises

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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) ${\\mathbb Z}_2 \\times {\\mathbb Z}_2 \\times {\\mathbb Z}_2\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $x^4 + x^2 + 1 = (x + 1)^2 (x + 2)^2 \\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
3

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

Hint

If

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

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 $\\alpha \\in \\gf(729)\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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) ${\\mathbb Q}(i)$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
6

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $S_3\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $K\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $S_n\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
10

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
11

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $G(E/F)\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
15

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $\\alpha = b^2 - 4ac\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
18

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $F[x]\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
20

We know that the cyclotomic polynomial

\n\\begin{equation*}\n\\Phi_p(x) = \\frac{x^p - 1}{x - 1} = x^{p - 1} + x^{p - 2} + \\cdots + x + 1\n\\end{equation*}\n

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

    \n\\begin{equation*}\n\\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},\n\\end{equation*}\n

    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 $G( {\\mathbb Q}( \\omega ) / {\\mathbb Q})\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $\\{ \\identity \\} \\subset G(E/L) \\subset G(E/K) \\subset G(E/F)\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $x^3 + x -3\\text{.}$

"]} ], "nbformat": 4, "nbformat_minor": 0, "metadata": {"kernelspec": {"display_name": "", "name": "sagemath"}, "language_info": {"codemirror_mode": {"name": "ipython", "version": 2}, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.8"}, "name": "exercises-galois.ipynb"} }