Abstract Algebra: Theory and Applications

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

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

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

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

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

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. $\mathbb{Q}(\sqrt{2},\sqrt[3]{2})$ 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 $\mathbb{Q}(\sqrt{3}+\sqrt{5})$

3.

Find the splitting field for each of the following polynomials.

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

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

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

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

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 $[F:\mathbb{Q}]=4\text{.}$

5.

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.

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 $\mathbb{Q}(\sqrt{3},\sqrt[4]{3},\sqrt[8]{3},\dots )$ 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 $[E:F]\le n!\text{.}$

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