Exercises21.5Exercises

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. $${\mathbb Q}( \sqrt{2}, \root 3 \of{2}\, )$$ 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. $$x^4 - 10 x^2 + 21$$ 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 $${\mathbb Z}_2[x] / \langle x^3 + x + 1 \rangle$$ 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 $$[F(\alpha) : F] = n\text{.}$$