{ "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":["

Section21.4Exercises

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
1

Show that each of the following numbers is algebraic over ${\\mathbb Q}$ by finding the minimal polynomial of the number over ${\\mathbb Q}\\text{.}$

  1. $\\sqrt{ 1/3 + \\sqrt{7} }$

  2. $\\sqrt{ 3} + \\sqrt[3]{5}$

  3. $\\sqrt{3} + \\sqrt{2}\\, i$

  4. $\\cos \\theta + i \\sin \\theta$ for $\\theta = 2 \\pi /n$ with $n \\in {\\mathbb N}$

  5. $\\sqrt{ \\sqrt[3]{2} - i }$

Hint

(a) $x^4 - (2/3) x^2 - 62/9\\text{;}$ (c) $x^4 - 2 x^2 + 25\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
2

Find a basis for each of the following field extensions. What is the degree of each extension?

  1. ${\\mathbb Q}( \\sqrt{3}, \\sqrt{6}\\, )$ over ${\\mathbb Q}$

  2. ${\\mathbb Q}( \\sqrt[3]{2}, \\sqrt[3]{3}\\, )$ over ${\\mathbb Q}$

  3. ${\\mathbb Q}( \\sqrt{2}, i)$ over ${\\mathbb Q}$

  4. ${\\mathbb Q}( \\sqrt{3}, \\sqrt{5}, \\sqrt{7}\\, )$ over ${\\mathbb Q}$

  5. ${\\mathbb Q}( \\sqrt{2}, \\root 3 \\of{2}\\, )$ over ${\\mathbb Q}$

  6. ${\\mathbb Q}( \\sqrt{8}\\, )$ over ${\\mathbb Q}(\\sqrt{2}\\, )$

  7. ${\\mathbb Q}(i, \\sqrt{2} +i, \\sqrt{3} + i )$ over ${\\mathbb Q}$

  8. ${\\mathbb Q}( \\sqrt{2} + \\sqrt{5}\\, )$ over ${\\mathbb Q} ( \\sqrt{5}\\, )$

  9. ${\\mathbb Q}( \\sqrt{2}, \\sqrt{6} + \\sqrt{10}\\, )$ over ${\\mathbb Q} ( \\sqrt{3} + \\sqrt{5}\\, )$

Hint

(a) $\\{ 1, \\sqrt{2}, \\sqrt{3}, \\sqrt{6}\\, \\}\\text{;}$ (c) $\\{ 1, i, \\sqrt{2}, \\sqrt{2}\\, i \\}\\text{;}$ (e) $\\{1, 2^{1/6}, 2^{1/3}, 2^{1/2}, 2^{2/3}, 2^{5/6} \\}\\text{.}$

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

Find the splitting field for each of the following polynomials.

  1. $x^4 - 10 x^2 + 21$ over ${\\mathbb Q}$

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

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

  4. $x^3 - 3$ over ${\\mathbb Q}$

Hint

(a) ${\\mathbb Q}(\\sqrt{3}, \\sqrt{7}\\, )\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
4

Consider the field extension ${\\mathbb Q}( \\sqrt[4]{3}, i )$ over $\\mathbb Q\\text{.}$

  1. Find a basis for the field extension ${\\mathbb Q}( \\sqrt[4]{3}, i )$ over $\\mathbb Q\\text{.}$ Conclude that $[{\\mathbb Q}( \\sqrt[4]{3}, i ): \\mathbb Q] = 8\\text{.}$

  2. Find all subfields $F$ of ${\\mathbb Q}( \\sqrt[4]{3}, i )$ such that $[F:\\mathbb Q] = 2\\text{.}$

  3. Find all subfields $F$ of ${\\mathbb Q}( \\sqrt[4]{3}, i )$ such that $[F:\\mathbb Q] = 4\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
5

Show that ${\\mathbb Z}_2[x] / \\langle x^3 + x + 1 \\rangle$ is a field with eight elements. Construct a multiplication table for the multiplicative group of the field.

Hint

Use the fact that the elements of ${\\mathbb Z}_2[x]/ \\langle x^3 + x + 1 \\rangle$ are 0, 1, $\\alpha\\text{,}$ $1 + \\alpha\\text{,}$ $\\alpha^2\\text{,}$ $1 + \\alpha^2\\text{,}$ $\\alpha + \\alpha^2\\text{,}$ $1 + \\alpha + \\alpha^2$ and the fact that $\\alpha^3 + \\alpha + 1 = 0\\text{.}$

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

Show that the regular 9-gon is not constructible with a straightedge and compass, but that the regular 20-gon is constructible.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
7

Prove that the cosine of one degree ($\\cos 1^\\circ$) is algebraic over ${\\mathbb Q}$ but not constructible.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
8

Can a cube be constructed with three times the volume of a given cube?

Hint

False.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
9

Prove that ${\\mathbb Q}(\\sqrt{3}, \\sqrt[4]{3}, \\sqrt[8]{3}, \\ldots )$ is an algebraic extension of ${\\mathbb Q}$ but not a finite extension.

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

Prove or disprove: $\\pi$ is algebraic over ${\\mathbb Q}(\\pi^3)\\text{.}$

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

Let $p(x)$ be a nonconstant polynomial of degree $n$ in $F[x]\\text{.}$ Prove that there exists a splitting field $E$ for $p(x)$ such that $[E : F] \\leq n!\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
12

Prove or disprove: ${\\mathbb Q}( \\sqrt{2}\\, ) \\cong {\\mathbb Q}( \\sqrt{3}\\, )\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
13

Prove that the fields ${\\mathbb Q}(\\sqrt[4]{3}\\, )$ and ${\\mathbb Q}(\\sqrt[4]{3}\\, i)$ are isomorphic but not equal.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
14

Let $K$ be an algebraic extension of $E\\text{,}$ and $E$ an algebraic extension of $F\\text{.}$ Prove that $K$ is algebraic over $F\\text{.}$ [Caution: Do not assume that the extensions are finite.]

Hint

Suppose that $E$ is algebraic over $F$ and $K$ is algebraic over $E\\text{.}$ Let $\\alpha \\in K\\text{.}$ It suffices to show that $\\alpha$ is algebraic over some finite extension of $F\\text{.}$ Since $\\alpha$ is algebraic over $E\\text{,}$ it must be the zero of some polynomial $p(x) = \\beta_0 + \\beta_1 x + \\cdots + \\beta_n x^n$ in $E[x]\\text{.}$ Hence $\\alpha$ is algebraic over $F(\\beta_0, \\ldots, \\beta_n)\\text{.}$

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

Prove or disprove: ${\\mathbb Z}[x] / \\langle x^3 -2 \\rangle$ is a field.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
16

Let $F$ be a field of characteristic $p\\text{.}$ Prove that $p(x) = x^p - a$ either is irreducible over $F$ or splits in $F\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
17

Let $E$ be the algebraic closure of a field $F\\text{.}$ Prove that every polynomial $p(x)$ in $F[x]$ splits in $E\\text{.}$

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

If every irreducible polynomial $p(x)$ in $F[x]$ is linear, show that $F$ is an algebraically closed field.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
19

Prove that if $\\alpha$ and $\\beta$ are constructible numbers such that $\\beta \\neq 0\\text{,}$ then so is $\\alpha / \\beta\\text{.}$

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

Show that the set of all elements in ${\\mathbb R}$ that are algebraic over ${\\mathbb Q}$ form a field extension of ${\\mathbb Q}$ that is not finite.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
21

Let $E$ be an algebraic extension of a field $F\\text{,}$ and let $\\sigma$ be an automorphism of $E$ leaving $F$ fixed. Let $\\alpha \\in E\\text{.}$ Show that $\\sigma$ induces a permutation of the set of all zeros of the minimal polynomial of $\\alpha$ that are in $E\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
22

Show that ${\\mathbb Q}( \\sqrt{3}, \\sqrt{7}\\, ) = {\\mathbb Q}( \\sqrt{3} + \\sqrt{7}\\, )\\text{.}$ Extend your proof to show that ${\\mathbb Q}( \\sqrt{a}, \\sqrt{b}\\, ) = {\\mathbb Q}( \\sqrt{a} + \\sqrt{b}\\, )\\text{,}$ where $\\gcd(a, b) = 1\\text{.}$

Hint

Since $\\{ 1, \\sqrt{3}, \\sqrt{7}, \\sqrt{21}\\, \\}$ is a basis for ${\\mathbb Q}( \\sqrt{3}, \\sqrt{7}\\, )$ over ${\\mathbb Q}\\text{,}$ ${\\mathbb Q}( \\sqrt{3}, \\sqrt{7}\\, ) \\supset {\\mathbb Q}( \\sqrt{3} +\\sqrt{7}\\, )\\text{.}$ Since $[{\\mathbb Q}( \\sqrt{3}, \\sqrt{7}\\, ) : {\\mathbb Q}] = 4\\text{,}$ $[{\\mathbb Q}( \\sqrt{3} + \\sqrt{7}\\, ) : {\\mathbb Q}] = 2$ or 4. Since the degree of the minimal polynomial of $\\sqrt{3} +\\sqrt{7}$ is 4, ${\\mathbb Q}( \\sqrt{3}, \\sqrt{7}\\, ) = {\\mathbb Q}( \\sqrt{3} +\\sqrt{7}\\, )\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
23

Let $E$ be a finite extension of a field $F\\text{.}$ If $[E:F] = 2\\text{,}$ show that $E$ is a splitting field of $F$ for some polynomial $f(x) \\in F[x]\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
24

Prove or disprove: Given a polynomial $p(x)$ in ${\\mathbb Z}_6[x]\\text{,}$ it is possible to construct a ring $R$ such that $p(x)$ has a root in $R\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
25

Let $E$ be a field extension of $F$ and $\\alpha \\in E\\text{.}$ Determine $[F(\\alpha): F(\\alpha^3)]\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
26

Let $\\alpha, \\beta$ be transcendental over ${\\mathbb Q}\\text{.}$ Prove that either $\\alpha \\beta$ or $\\alpha + \\beta$ is also transcendental.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
27

Let $E$ be an extension field of $F$ and $\\alpha \\in E$ be transcendental over $F\\text{.}$ Prove that every element in $F(\\alpha)$ that is not in $F$ is also transcendental over $F\\text{.}$

Hint

Let $\\beta \\in F(\\alpha)$ not in $F\\text{.}$ Then $\\beta = p(\\alpha)/q(\\alpha)\\text{,}$ where $p$ and $q$ are polynomials in $\\alpha$ with $q(\\alpha) \\neq 0$ and coefficients in $F\\text{.}$ If $\\beta$ is algebraic over $F\\text{,}$ then there exists a polynomial $f(x) \\in F[x]$ such that $f(\\beta) = 0\\text{.}$ Let $f(x) = a_0 + a_1 x + \\cdots + a_n x^n\\text{.}$ Then

\n\\begin{equation*}\n0 = f(\\beta) = f\\left( \\frac{p(\\alpha)}{q(\\alpha)} \\right) = a_0 + a_1 \\left( \\frac{p(\\alpha)}{q(\\alpha)} \\right) + \\cdots + a_n \\left( \\frac{p(\\alpha)}{q(\\alpha)} \\right)^n.\n\\end{equation*}\n

Now multiply both sides by $q(\\alpha)^n$ to show that there is a polynomial in $F[x]$ that has $\\alpha$ as a zero.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
28

Let $\\alpha$ be a root of an irreducible monic polynomial $p(x) \\in F[x]\\text{,}$ with $\\deg p = n\\text{.}$ Prove that $[F(\\alpha) : F] = n\\text{.}$

Hint

See the comments following Theorem 21.13.

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