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Exercises 21.5 Exercises

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. \(\displaystyle \sqrt{ \sqrt[3]{2} - i }\)
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. \({\mathbb Q}( \sqrt{2}, \root 3 \of{2}\, )\) 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 \({\mathbb Q} ( \sqrt{3} + \sqrt{5}\, )\)
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 \(a \neq b\) and neither \(a\) nor \(b\) is a perfect square.
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\text{.} \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 21.1.13.
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