Abstract Algebra: Theory and Applications

Show that ${\mathbb{Z}}_2[x]/⟨x^3+x+1⟩$ 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 $\mathbb{Q}(\sqrt{3},\sqrt[4]{3},\sqrt[8]{3},\dots )$ 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 $[E:F]\le n!\text{.}$

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: $\mathbb{Z}[x]/⟨x^3-2⟩$ 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 $\beta \ne 0\text{,}$ 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 $a\ne b$ 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 $\mathrm{deg}p=n\text{.}$ Prove that $[F(\alpha ):F]=n\text{.}$