Fields: exercises

Practice (28)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Show that each of the following numbers is algebraic over \({\mathbb Q}\) by finding the minimal polynomial of the number over \({\mathbb Q}\).

   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. \(\sqrt{ \sqrt[3]{2} - i }\)

   كشفت الإجابة

   Hint:

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

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 \({\mathbb Q} ( \sqrt{3} + \sqrt{5}\, )\)

   كشفت الإجابة

   Hint:

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

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 \({\mathbb Q}\)

   كشفت الإجابة

   Hint:

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

4. Consider the field extension \({\mathbb Q}( \sqrt[4]{3}, i )\) over \(\mathbb Q\).

   1. Find a basis for the field extension \({\mathbb Q}( \sqrt[4]{3}, i )\) over \(\mathbb Q\). Conclude that \([{\mathbb Q}( \sqrt[4]{3}, i ): \mathbb Q] = 8\).

   2. Find all subfields \(F\) of \({\mathbb Q}( \sqrt[4]{3}, i )\) such that \([F:\mathbb Q] = 2\).

   3. Find all subfields \(F\) of \({\mathbb Q}( \sqrt[4]{3}, i )\) such that \([F:\mathbb Q] = 4\).

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.

   كشفت الإجابة

   Hint:

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

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?

   كشفت الإجابة

   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 \({\mathbb Q}(\pi^3)\).

11. Let \(p(x)\) be a nonconstant polynomial of degree \(n\) in \(F[x]\). Prove that there exists a splitting field \(E\) for \(p(x)\) such that \([E : F] \leq n!\).

12. Prove or disprove: \({\mathbb Q}( \sqrt{2}\, ) \cong {\mathbb Q}( \sqrt{3}\, )\).

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 \(E\), and \(E\) an algebraic extension of \(F\). Prove that \(K\) is algebraic over \(F\). [ Caution: Do not assume that the extensions are finite.]

    كشفت الإجابة

    Hint:

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

15. Prove or disprove: \({\mathbb Z}[x] / \langle x^3 -2 \rangle\) is a field.

16. Let \(F\) be a field of characteristic \(p\). Prove that \(p(x) = x^p - a\) either is irreducible over \(F\) or splits in \(F\).

17. Let \(E\) be the algebraic closure of a field \(F\). Prove that every polynomial \(p(x)\) in \(F[x]\) splits in \(E\).

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 \(\beta \neq 0\), then so is \(\alpha / \beta\).

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 \(F\), and let \(\sigma\) be an automorphism of \(E\) leaving \(F\) fixed. Let \(\alpha \in E\). Show that \(\sigma\) induces a permutation of the set of all zeros of the minimal polynomial of \(\alpha\) that are in \(E\).

22. Show that \({\mathbb Q}( \sqrt{3}, \sqrt{7}\, ) = {\mathbb Q}( \sqrt{3} + \sqrt{7}\, )\). Extend your proof to show that \({\mathbb Q}( \sqrt{a}, \sqrt{b}\, ) = {\mathbb Q}( \sqrt{a} + \sqrt{b}\, )\), where \(a \neq b\) and neither \(a\) nor \(b\) is a perfect square.

    كشفت الإجابة

    Hint:

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

23. Let \(E\) be a finite extension of a field \(F\). If \([E:F] = 2\), show that \(E\) is a splitting field of \(F\) for some polynomial \(f(x) \in F[x]\).

24. Prove or disprove: Given a polynomial \(p(x)\) in \({\mathbb Z}_6[x]\), it is possible to construct a ring \(R\) such that \(p(x)\) has a root in \(R\).

25. Let \(E\) be a field extension of \(F\) and \(\alpha \in E\). Determine \([F(\alpha): F(\alpha^3)]\).

26. Let \(\alpha, \beta\) be transcendental over \({\mathbb Q}\). 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 \(F\). Prove that every element in \(F(\alpha)\) that is not in \(F\) is also transcendental over \(F\).

    كشفت الإجابة

    Hint:

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

    كشفت الإجابة

    Hint:

    See the comments following .

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

\theta

theta
The usual name for an angle.

\sin,\ \cos,\ \tan

sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.

x \in A,\ A \subseteq B

element of, subset
x belongs to A; every element of A is in B.

i

imaginary unit
i² = −1.

^\circ

degrees
1/360 of a full turn. 180° = π radians.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\neq

not equal
The two sides are different.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

\blacksquare\ \text{or}\ \square

end of proof (halmos)
Marks the point where the statement has been established.

a \equiv b \pmod n

congruent modulo n
n divides a − b; a and b have the same remainder.

a \mid b,\ \gcd(a,b)

divides, greatest common divisor
b is a multiple of a; the largest number dividing both.

(G, \cdot),\ e,\ g^{-1}

group, identity, inverse
A set with an operation; the do-nothing element; the element that undoes g.

G \cong H,\ G / N

isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.

\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n

integers modulo n
The remainders 0…n−1 with clock arithmetic.

Questions people ask

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group — that is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

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x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.

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