MODERN ALGEBRA (MATH 210) EXAM PRACTICE PROBLEMS AND GALOIS THEORY PRACTICE PROBLEMS

The final exam will have roughly 9 problems, with roughly 2 on groups, 2 on rings, 1 on Jordan-Holder, 1 on semidirect products, and 3 on Galois theory.

1. Practice problems for the exam

P1. Show that the rotation group of the cube is isomorphic to S_{4}.

P2. If A and B are subgroups of finite index in a group G, and the indices of A and B in G are relatively prime, show that G = AB.

P3. Suppose p is an odd prime.

(a) Show that exactly half of (ℤ/pℤ)^{∗} = {1, 2, …, p − 1} are squares modulo p. (Hint: consider the structure of the group (ℤ/pℤ)^{∗}.)

(b) Prove that a^{(p−1)/2} ≡ ±1 (mod p) for all a ∈ (ℤ/pℤ)^{∗}.

(c) Show that a^{(p−1)/2} ≡ 1 (mod p) if and only if a is a perfect square in (ℤ/pℤ)^{∗}.

(d) Show that if neither a nor b are perfect squares modulo p, then ab is a perfect square modulo p.

P4. Are the following ideals prime in ℂ[x, y]?

(a) (x, y − 1),

(b) (x, y^{2}),

(c) (y − x^{2}, y − 1).

P5. If α ∈ E an extension of F, and f(x), g(x) ∈ F [x] with f(α) = g(α) = 0, and f is irreducible, show that f(x) is a factor of g(x).

P6. Let p and q be distinct primes.

(i) Prove that every group of order pq is solvable. (Hint: show that the group can’t be simple.)

(ii) Prove that every group G of order p^{2}q is solvable. (Hint: Show that G can’t be simple.)

P7. Suppose f(x) ∈ F [x^{p}] is irreducible of positive degree. Show that f(x) is not separable.

Date: Thursday, December 5, 2002.

P8.

(a) Suppose the splitting field E of a cubic f(x) over ℚ has Galois group S_{3}, and f(x) has roots x_{1}, x_{2}, x_{3}. Show that E^{A_{3}} is generated (over ℚ) by Δ = (x_{1}−x_{2})(x_{1}−x_{3})(x_{2}−x_{3}).

(b) Show that the splitting field E of a cubic f(x) = x^{3} + ax^{2} + bx + c over ℚ has Galois group ℤ/3ℤ if and only if \sqrt{Δ} ∈ ℚ, where Δ := a^{2}b^{2}−4b^{3}−4a^{3}c−27c^{2}+18abc. (If E has roots x_{1}, x_{2}, x_{3}, then you may use the fact that [(x_{1}−x_{2})(x_{1}−x_{3})(x_{2}−x_{3})]^{2} = Δ, which you could have computed with some effort from a problem on the last problem set.)

P9. Let E be the splitting field of

f(x) = (x^{7} − 1)/(x − 1) = x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x^{1} + 1

over ℚ. Let ζ be a zero of f(x), i.e. a primitive seventh root of 1.

(a) Show that f(x) is irreducible. (Hint: consider f(y + 1) and use Eisenstein.)

(b) Find the degree of the extension E/ℚ.

(c) Show that the Galois group of E/ℚ is cyclic, and find an explicit generator.

(d) Let β = ζ + ζ^{2} + ζ^{4}. Show that the intermediate field ℚ(β) is actually ℚ(\sqrt{−7}). (Hint: first show that [ℚ(β) : ℚ] = 2 by finding a linear dependence over ℚ among {1, β, β^{2}}.)

(e) Let γ_{q} = ζ + ζ^{q}. Find a q such that ℚ(γ_{q}) is a degree 3 extension of ℚ. (Possible hint: use (c).) Is this extension Galois?

2. Galois theory practice problems

G1. Suppose G is the Galois group of the Galois extension E/F, and H < G is a subgroup (not necessarily normal) with α conjugates. Find | Aut(H/k)| in terms of α, |G|, |H|.

G2. Describe the splitting field of x^{3} − 2 over ℚ, and all intermediate fields.

G3. Suppose F is characteristic 2, and E/F is a degree 2 extension with E = F (α), and α satisfying T^{2} + bT + c = 0. What is the condition on b and c such that E/F is Galois?

G4. Show that the polynomial t^{5} − 4t + 2 is not soluble by radicals.