Chapter 21

Show that each of the following numbers is algebraic over \(Q\) by finding the minimal polynomial of the number over \(\mathbb{Q}\). (a) \(\sqrt{1 / 3+\sqrt{7}}\) (b) \(\sqrt{3}+\sqrt[3]{5}\) (c) \(\sqrt{3}+\sqrt{2} i\) (d) \(\cos \theta+i \sin \theta\) for \(\theta=2 \pi / n\) with \(n \in \mathbb{N}\) (e) \(\sqrt{\sqrt[3]{2}-i}\)

Find a basis for each of the following field extensions. What is the degree of each extension? (a) \(Q(\sqrt{3}, \sqrt{6})\) over \(\mathbb{Q}\) (b) \(\mathbb{Q}(\sqrt[3]{2}, \sqrt[3]{3})\) over \(\mathbf{Q}\) (c) \(Q(\sqrt{2}, i)\) over \(Q\) (d) \(Q(\sqrt{3}, \sqrt{5}, \sqrt{7})\) over \(\mathbb{Q}\) (e) \(\mathrm{Q}(\sqrt{2}, \sqrt[3]{2})\) over \(\mathrm{Q}\) (f) \(Q(\sqrt{8})\) over \(Q(\sqrt{2})\) (g) \(\mathbb{Q}(i, \sqrt{2}+i, \sqrt{3}+i)\) over \(\mathbb{Q}\) (h) \(Q(\sqrt{2}+\sqrt{5})\) over \(Q(\sqrt{5})\) (i) \(\mathbb{Q}(\sqrt{2}, \sqrt{6}+\sqrt{10})\) over \(Q(\sqrt{3}+\sqrt{5})\)

Find the splitting field for each of the following polynomials. (a) \(x^{4}-10 x^{2}+21\) over \(\mathbb{Q}\) (c) \(x^{3}+2 x+2\) over \(\mathbb{Z}_{3}\) (b) \(x^{4}+1\) over \(\mathbb{Q}\) (d) \(x^{3}-3\) over \(Q\)

Consider the field extension \(Q(\sqrt[4]{3}, i)\) over \(\mathbb{Q}\). (a) Find a basis for the field extension \(\mathbb{Q}(\sqrt[4]{3}, i)\) over \(\mathbf{Q}\). Conclude that \([\mathbb{Q}(\sqrt[4]{3}, i): \mathbb{Q}]=8\). (b) Find all subfields \(F\) of \(\mathbb{Q}(\sqrt[4]{3}, i)\) such that \([F: \mathbb{Q}]=2\). (c) Find all subfields \(F\) of \(\mathbb{Q}(\sqrt[4]{3}, i)\) such that \([F: \mathbb{Q}]=4\).

Show that \(\mathbb{Z}_{2}[x] /\left\langle x^{3}+x+1\right)\) is a field with eight elements. Construct a multiplication table for the multiplicative group of the field.

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

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

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

Prove that \(\mathbb{Q}(\sqrt{3}, \sqrt[4]{3}, \sqrt[5]{3}, \ldots)\) is an algebraic extension of \(Q\) but not a finite extension. v

Prove or disprove: \(\pi\) is algebraic over \(Q\left(\pi^{3}\right)\).

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

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

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

Prove or disprove: \(\mathbb{Z}[x] /\left(x^{3}-2\right)\) is a field.

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

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

Prove that if \(\alpha\) and \(\beta\) are constructible numbers such that \(\beta \neq 0\), then so is \(\alpha / \beta\).

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

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

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

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

Prove or disprove: Given a polynomial \(p(x)\) in \(\mathbb{Z}_{6}[x],\) it is possible to construct a ring \(R\) such that \(p(x)\) has a root in \(R\).

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

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

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

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