exercises-fields.ipynb

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Section21.4Exercises

¶

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

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

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

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

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

Hint

(a) $x^4 - (2/3) x^2 - 62/9\text{;}$ (c) $x^4 - 2 x^2 + 25\text{.}$

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. ParseError: KaTeX parse error: Undefined control sequence: \root at position 24: … Q}( \sqrt{2}, \̲r̲o̲o̲t̲ ̲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 ${\mathbb Q} ( \sqrt{3} + \sqrt{5}\, )$

Hint

(a) $\{ 1, \sqrt{2}, \sqrt{3}, \sqrt{6}\, \}\text{;}$ (c) $\{ 1, i, \sqrt{2}, \sqrt{2}\, i \}\text{;}$ (e) $\{1, 2^{1/6}, 2^{1/3}, 2^{1/2}, 2^{2/3}, 2^{5/6} \}\text{.}$

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 ${\mathbb Q}$

Hint

(a) ${\mathbb Q}(\sqrt{3}, \sqrt{7}\, )\text{.}$

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.

Hint

Use the fact that the elements of ${\mathbb Z}_2[x]/ \langle x^3 + x + 1 \rangle$ are 0, 1, $\alpha\text{,}$ $1 + \alpha\text{,}$ $\alpha^2\text{,}$ $1 + \alpha^2\text{,}$ $\alpha + \alpha^2\text{,}$ $1 + \alpha + \alpha^2$ and the fact that $\alpha^3 + \alpha + 1 = 0\text{.}$

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?

Hint

False.

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.]

Hint

Suppose that $E$ is algebraic over $F$ and $K$ is algebraic over $E\text{.}$ Let $\alpha \in K\text{.}$ It suffices to show that $\alpha$ is algebraic over some finite extension of $F\text{.}$ Since $\alpha$ is algebraic over $E\text{,}$ it must be the zero of some polynomial $p(x) = \beta_0 + \beta_1 x + \cdots + \beta_n x^n$ in $E[x]\text{.}$ Hence $\alpha$ is algebraic over $F(\beta_0, \ldots, \beta_n)\text{.}$

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 $\gcd(a, b) = 1\text{.}$

Hint

Since $\{ 1, \sqrt{3}, \sqrt{7}, \sqrt{21}\, \}$ is a basis for ${\mathbb Q}( \sqrt{3}, \sqrt{7}\, )$ over ${\mathbb Q}\text{,}$ ${\mathbb Q}( \sqrt{3}, \sqrt{7}\, ) \supset {\mathbb Q}( \sqrt{3} +\sqrt{7}\, )\text{.}$ Since $[{\mathbb Q}( \sqrt{3}, \sqrt{7}\, ) : {\mathbb Q}] = 4\text{,}$ $[{\mathbb Q}( \sqrt{3} + \sqrt{7}\, ) : {\mathbb Q}] = 2$ or 4. Since the degree of the minimal polynomial of $\sqrt{3} +\sqrt{7}$ is 4, ${\mathbb Q}( \sqrt{3}, \sqrt{7}\, ) = {\mathbb Q}( \sqrt{3} +\sqrt{7}\, )\text{.}$

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 $F\text{.}$

Hint

Let $\beta \in F(\alpha)$ not in $F\text{.}$ Then $\beta = p(\alpha)/q(\alpha)\text{,}$ where $p$ and $q$ are polynomials in $\alpha$ with $q(\alpha) \neq 0$ and coefficients in $F\text{.}$ If $\beta$ is algebraic over $F\text{,}$ then there exists a polynomial $f(x) \in F[x]$ such that $f(\beta) = 0\text{.}$ Let $f(x) = a_0 + a_1 x + \cdots + a_n x^n\text{.}$ Then

$$\begin{equation*} 0 = f(\beta) = f\left( \frac{p(\alpha)}{q(\alpha)} \right) = a_0 + a_1 \left( \frac{p(\alpha)}{q(\alpha)} \right) + \cdots + a_n \left( \frac{p(\alpha)}{q(\alpha)} \right)^n. \end{equation*}$$

Now multiply both sides by $q(\alpha)^n$ to show that there is a polynomial in $F[x]$ that has $\alpha$ as a zero.

28

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

Hint

See the comments following Theorem 21.13.