Exercises 21.5 Exercises
1.
Show that each of the following numbers is algebraic over \({\mathbb Q}\) by finding the minimal polynomial of the number over \({\mathbb Q}\text{.}\)
\(\displaystyle \sqrt{ 1/3 + \sqrt{7} }\)
\(\displaystyle \sqrt{ 3} + \sqrt[3]{5}\)
\(\displaystyle \sqrt{3} + \sqrt{2}\, i\)
\(\cos \theta + i \sin \theta\) for \(\theta = 2 \pi /n\) with \(n \in {\mathbb N}\)
\(\displaystyle \sqrt{ \sqrt[3]{2} - i }\)
2.
Find a basis for each of the following field extensions. What is the degree of each extension?
\({\mathbb Q}( \sqrt{3}, \sqrt{6}\, )\) over \({\mathbb Q}\)
\({\mathbb Q}( \sqrt[3]{2}, \sqrt[3]{3}\, )\) over \({\mathbb Q}\)
\({\mathbb Q}( \sqrt{2}, i)\) over \({\mathbb Q}\)
\({\mathbb Q}( \sqrt{3}, \sqrt{5}, \sqrt{7}\, )\) over \({\mathbb Q}\)
\({\mathbb Q}( \sqrt{2}, \root 3 \of{2}\, )\) over \({\mathbb Q}\)
\({\mathbb Q}( \sqrt{8}\, )\) over \({\mathbb Q}(\sqrt{2}\, )\)
\({\mathbb Q}(i, \sqrt{2} +i, \sqrt{3} + i )\) over \({\mathbb Q}\)
\({\mathbb Q}( \sqrt{2} + \sqrt{5}\, )\) over \({\mathbb Q} ( \sqrt{5}\, )\)
\({\mathbb Q}( \sqrt{2}, \sqrt{6} + \sqrt{10}\, )\) over \({\mathbb Q} ( \sqrt{3} + \sqrt{5}\, )\)
3.
Find the splitting field for each of the following polynomials.
\(x^4 - 10 x^2 + 21\) over \({\mathbb Q}\)
\(x^4 + 1\) over \({\mathbb Q}\)
\(x^3 + 2x + 2\) over \({\mathbb Z}_3\)
\(x^3 - 3\) over \({\mathbb Q}\)
4.
Consider the field extension \({\mathbb Q}( \sqrt[4]{3}, i )\) over \(\mathbb Q\text{.}\)
Find a basis for the field extension \({\mathbb Q}( \sqrt[4]{3}, i )\) over \(\mathbb Q\text{.}\) Conclude that \([{\mathbb Q}( \sqrt[4]{3}, i ): \mathbb Q] = 8\text{.}\)
Find all subfields \(F\) of \({\mathbb Q}( \sqrt[4]{3}, i )\) such that \([F:\mathbb Q] = 2\text{.}\)
Find all subfields \(F\) of \({\mathbb Q}( \sqrt[4]{3}, i )\) such that \([F:\mathbb Q] = 4\text{.}\)
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.
6.
Show that the regular \(9\)-gon is not constructible with a straightedge and compass, but that the regular \(20\)-gon is constructible.
7.
Prove that the cosine of one degree (\(\cos 1^\circ\)) is algebraic over \({\mathbb Q}\) but not constructible.
8.
Can a cube be constructed with three times the volume of a given cube?
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.
10.
Prove or disprove: \(\pi\) is algebraic over \({\mathbb Q}(\pi^3)\text{.}\)
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 \([E : F] \leq n!\text{.}\)
12.
Prove or disprove: \({\mathbb Q}( \sqrt{2}\, ) \cong {\mathbb Q}( \sqrt{3}\, )\text{.}\)
13.
Prove that the fields \({\mathbb Q}(\sqrt[4]{3}\, )\) and \({\mathbb Q}(\sqrt[4]{3}\, i)\) are isomorphic but not equal.
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.]
15.
Prove or disprove: \({\mathbb Z}[x] / \langle x^3 -2 \rangle\) is a field.
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 \(F\text{.}\)
17.
Let \(E\) be the algebraic closure of a field \(F\text{.}\) Prove that every polynomial \(p(x)\) in \(F[x]\) splits in \(E\text{.}\)
18.
If every irreducible polynomial \(p(x)\) in \(F[x]\) is linear, show that \(F\) is an algebraically closed field.
19.
Prove that if \(\alpha\) and \(\beta\) are constructible numbers such that \(\beta \neq 0\text{,}\) then so is \(\alpha / \beta\text{.}\)
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.
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 \(E\text{.}\)
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 \(a \neq b\) and neither \(a\) nor \(b\) is a perfect square.
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 \(f(x) \in F[x]\text{.}\)
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 \(R\text{.}\)
25.
Let \(E\) be a field extension of \(F\) and \(\alpha \in E\text{.}\) Determine \([F(\alpha): F(\alpha^3)]\text{.}\)
26.
Let \(\alpha, \beta\) be transcendental over \({\mathbb Q}\text{.}\) Prove that either \(\alpha \beta\) or \(\alpha + \beta\) is also transcendental.
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 \(F\text{.}\)
28.
Let \(\alpha\) be a root of an irreducible monic polynomial \(p(x) \in F[x]\text{,}\) with \(\deg p = n\text{.}\) Prove that \([F(\alpha) : F] = n\text{.}\)