Abstract Algebra Theory and Applications

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.

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 (\(\cos 1^\circ\)) is algebraic over \({\mathbb Q}\) but not constructible.

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

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

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

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 \([E : F] \leq n!\text{.}\)

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

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

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.]

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

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

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

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

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

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.

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 \(E\text{.}\)

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.

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 \(f(x) \in F[x]\text{.}\)

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 \(R\text{.}\)

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

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

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 \(F\text{.}\)

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