Galois Theory: exercises

Practice (22)

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

1. Compute each of the following Galois groups. Which of these field extensions are normal field extensions? If the extension is not normal, find a normal extension of \({\mathbb Q}\) in which the extension field is contained.

   1. \(G({\mathbb Q}(\sqrt{30}\, ) / {\mathbb Q})\)

   2. \(G({\mathbb Q}(\sqrt[4]{5}\, ) / {\mathbb Q})\)

   3. \(G( {\mathbb Q}(\sqrt{2}, \sqrt{3}, \sqrt{5}\, )/ {\mathbb Q} )\)

   4. \(G({\mathbb Q}(\sqrt{2}, \sqrt[3]{2}, i) / {\mathbb Q})\)

   5. \(G({\mathbb Q}(\sqrt{6}, i) / {\mathbb Q})\)

   Откройте ответ.

   Hint:

   (a) \({\mathbb Z}_2\); (c) \({\mathbb Z}_2 \times {\mathbb Z}_2 \times {\mathbb Z}_2\).

2. Determine the separability of each of the following polynomials.

   1. \(x^3 + 2 x^2 - x - 2\) over \({\mathbb Q}\)

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

   3. \(x^4 + x^2 + 1\) over \({\mathbb Z}_3\)

   4. \(x^3 +x^2 + 1\) over \({\mathbb Z}_2\)

   Откройте ответ.

   Hint:

   (a) Separable over \(\mathbb Q\) since \(x^3 + 2 x^2 - x - 2 = (x - 1)(x + 1)(x + 2)\); (c) not separable over \(\mathbb Z_3\) since \(x^4 + x^2 + 1 = (x + 1)^2 (x + 2)^2\).

3. Give the order and describe a generator of the Galois group of \(\gf(729)\) over \(\gf(9)\).

   Откройте ответ.

   Hint:

   If \[\begin{aligned}\end{aligned}\], then \(G(\gf(729)/ \gf(9)) \cong {\mathbb Z}_3\). A generator for \(G(\gf(729)/ \gf(9))\) is \(\sigma\), where \(\sigma_{3^6}( \alpha) = \alpha^{3^6} = \alpha^{729}\) for \(\alpha \in \gf(729)\).

4. Determine the Galois groups of each of the following polynomials in \({\mathbb Q}[x]\); hence, determine the solvability by radicals of each of the polynomials.

   1. \(x^5 - 12 x^2 + 2\)

   2. \(x^5 - 4 x^4 + 2 x + 2\)

   3. \(x^3 - 5\)

   4. \(x^4 - x^2 - 6\)

   5. \(x^5 + 1\)

   6. \((x^2 - 2)(x^2 + 2)\)

   7. \(x^8 - 1\)

   8. \(x^8 + 1\)

   9. \(x^4 - 3 x^2 -10\)

   Откройте ответ.

   Hint:

   (a) \(S_5\); (c) \(S_3\); (g) see .

5. Find a primitive element in the splitting field of each of the following polynomials in \({\mathbb Q}[x]\).

   1. \(x^4 - 1\)

   2. \(x^4 - 8 x^2 + 15\)

   3. \(x^4 - 2 x^2 - 15\)

   4. \(x^3 - 2\)

   Откройте ответ.

   Hint:

   (a) \({\mathbb Q}(i)\)

6. Prove that the Galois group of an irreducible quadratic polynomial is isomorphic to \({\mathbb Z}_2\).

7. Prove that the Galois group of an irreducible cubic polynomial is isomorphic to \(S_3\) or \({\mathbb Z}_3\).

   Откройте ответ.

   Hint:

   Let \(E\) be the splitting field of a cubic polynomial in \(F[x]\). Show that \([E:F]\) is less than or equal to \(6\) and is divisible by \(3\). Since \(G(E/F)\) is a subgroup of \(S_3\) whose order is divisible by \(3\), conclude that this group must be isomorphic to \({\mathbb Z}_3\) or \(S_3\).

8. Let \(F \subset K \subset E\) be fields. If \(E\) is a normal extension of \(F\), show that \(E\) must also be a normal extension of \(K\).

9. Let \(G\) be the Galois group of a polynomial of degree \(n\). Prove that \(|G|\) divides \(n!\).

   Откройте ответ.

   Hint:

   \(G\) is a subgroup of \(S_n\).

10. Let \(F \subset E\). If \(f(x)\) is solvable over \(F\), show that \(f(x)\) is also solvable over \(E\).

11. Construct a polynomial \(f(x)\) in \({\mathbb Q}[x]\) of degree \(7\) that is not solvable by radicals.

12. Let \(p\) be prime. Prove that there exists a polynomial \(f(x) \in{\mathbb Q}[x]\) of degree \(p\) with Galois group isomorphic to \(S_p\). Conclude that for each prime \(p\) with \(p \geq 5\) there exists a polynomial of degree \(p\) that is not solvable by radicals.

13. Let \(p\) be a prime and \({\mathbb Z}_p(t)\) be the field of rational functions over \({\mathbb Z}_p\). Prove that \(f(x) = x^p - t\) is an irreducible polynomial in \({\mathbb Z}_p(t)[x]\). Show that \(f(x)\) is not separable.

14. Let \(E\) be an extension field of \(F\). Suppose that \(K\) and \(L\) are two intermediate fields. If there exists an element \(\sigma \in G(E/F)\) such that \(\sigma(K) = L\), then \(K\) and \(L\) are said to be conjugate fields. Prove that \(K\) and \(L\) are conjugate if and only if \(G(E/K)\) and \(G(E/L)\) are conjugate subgroups of \(G(E/F)\).

15. Let \(\sigma \in \aut( {\mathbb R} )\). If \(a\) is a positive real number, show that \(\sigma( a) > 0\).

16. Let \(K\) be the splitting field of \(x^3 + x^2 + 1 \in {\mathbb Z}_2[x]\). Prove or disprove that \(K\) is an extension by radicals.

    Откройте ответ.

    Hint:

    True.

17. Let \(F\) be a field such that \(\chr(F) \neq 2\). Prove that the splitting field of \(f(x) = a x^2 + b x + c\) is \(F( \sqrt{\alpha}\, )\), where \(\alpha = b^2 - 4ac\).

18. Prove or disprove: Two different subgroups of a Galois group will have different fixed fields.

19. Let \(K\) be the splitting field of a polynomial over \(F\). If \(E\) is a field extension of \(F\) contained in \(K\) and \([E:F] = 2\), then \(E\) is the splitting field of some polynomial in \(F[x]\).

20. We know that the cyclotomic polynomial \[\begin{aligned}\end{aligned}\] is irreducible over \({\mathbb Q}\) for every prime \(p\). Let \(\omega\) be a zero of \(\Phi_p(x)\), and consider the field \({\mathbb Q}(\omega)\).

    1. Show that \(\omega, \omega^2, \ldots, \omega^{p-1}\) are distinct zeros of \(\Phi_p(x)\), and conclude that they are all the zeros of \(\Phi_p(x)\).

    2. Show that \(G( {\mathbb Q}( \omega ) / {\mathbb Q} )\) is abelian of order \(p - 1\).

    3. Show that the fixed field of \(G( {\mathbb Q}( \omega ) / {\mathbb Q} )\) is \({\mathbb Q}\).

    Откройте ответ.

    Hint:

    1. Clearly \(\omega, \omega^2, \ldots, \omega^{p - 1}\) are distinct since \(\omega \neq 1\) or 0. To show that \(\omega^i\) is a zero of \(\Phi_p\), calculate \(\Phi_p( \omega^i)\).

    2. The conjugates of \(\omega\) are \(\omega, \omega^2, \ldots, \omega^{p - 1}\). Define a map \(\phi_i: {\mathbb Q}(\omega) \rightarrow {\mathbb Q}(\omega^i)\) by \[\begin{aligned}\end{aligned}\], where \(a_i \in {\mathbb Q}\). Prove that \(\phi_i\) is an isomorphism of fields. Show that \(\phi_2\) generates \(G({\mathbb Q}(\omega)/{\mathbb Q})\).

    3. Show that \(\{ \omega, \omega^2, \ldots, \omega^{p - 1} \}\) is a basis for \({\mathbb Q}( \omega )\) over \({\mathbb Q}\), and consider which linear combinations of \(\omega, \omega^2, \ldots, \omega^{p - 1}\) are left fixed by all elements of \(G( {\mathbb Q}( \omega ) / {\mathbb Q})\).

21. Let \(F\) be a finite field or a field of characteristic zero. Let \(E\) be a finite normal extension of \(F\) with Galois group \(G(E/F)\). Prove that \(F \subset K \subset L \subset E\) if and only if \(\{ \identity \} \subset G(E/L) \subset G(E/K) \subset G(E/F)\).

22. Let \(F\) be a field of characteristic zero and let \(f(x) \in F[x]\) be a separable polynomial of degree \(n\). If \(E\) is the splitting field of \(f(x)\), let \(\alpha_1, \ldots, \alpha_n\) be the roots of \(f(x)\) in \(E\). Let \(\Delta = \prod_{i \lt j} (\alpha_i - \alpha_j)\). We define the discriminant of \(f(x)\) to be \(\Delta^2\). \(\Delta^2\) discriminant of a polynomial

    1. If \(f(x) = x^2 + b x + c\), show that \(\Delta^2 = b^2 - 4c\).

    2. If \(f(x) = x^3 + p x + q\), show that \(\Delta^2 = - 4p^3 - 27q^2\).

    3. Prove that \(\Delta^2\) is in \(F\).

    4. If \(\sigma \in G(E/F)\) is a transposition of two roots of \(f(x)\), show that \(\sigma( \Delta ) = -\Delta\).

    5. If \(\sigma \in G(E/F)\) is an even permutation of the roots of \(f(x)\), show that \(\sigma( \Delta ) = \Delta\).

    6. Prove that \(G(E/F)\) is isomorphic to a subgroup of \(A_n\) if and only if \(\Delta \in F\).

    7. Determine the Galois groups of \(x^3 + 2 x - 4\) and \(x^3 + x -3\).

Symbols used here

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

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

x \in A,\ A \subseteq B

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

\sigma,\ s,\ \sigma^2

standard deviation, sample s.d., variance
Typical distance from the mean; its square.

i

imaginary unit
i² = −1.

\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.

\operatorname{Hom}(A, B),\ f \circ g

arrows from A to B, composition
The set of morphisms; do g then f.

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.

Попробуй сам.

⌨ Type ✎ Draw

Проблема с математикой.

Решение

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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