Algebra Prelim, January 2019

Do all problems

1. Prove that a group of order 992 = 31·32 has either a normal subgroup of order 32 or a normal subgroup of order 62.

2. Let R be a commutative ring with a 1≠ 0. Prove that the set of prime ideals of R has a minimal element with respect to inclusion.

3. Let R be a commutative ring with a 1≠ 0 and suppose every irreducible R-module is free. Prove that R is a field. Hint: show that if R≌R⊕M, then M = 0.

4. Let k be a field and let M be a finitely generated k[x]-module. Prove that there exists a submodule N of M with N≠M and N≌M if and only if dim_kM = ∞.

5. Let p be a prime, let f∈ℚ[x] be an irreducible polynomial, and let α,β be distinct roots of f. Suppose that ℚ(α) = ℚ(β) and that [ℚ(α) : ℚ] = p. Prove that ℚ(α) is a Galois extension of ℚ.

6. Let p be a prime, let n∈ℕ, let k = 𝔽_p, and let K = 𝔽_pⁿ. Define θ : K→K by θ(a) = a^p ( 𝔽_pⁿ denotes the field with pⁿ elements).

   (a)

   Prove that θ is a k-linear map.

   (b)

   Prove that the minimal polynomial of θ (as a k-linear map) divides
   Xⁿ - 1.

   (c)

   Prove that if n divides p - 1, then θ is diagonalizable over k (i.e. there is a k-basis of K such that the matrix of θ with respect to this basis is a diagonal matrix).

7. Compute the character table of S₃×ℤ/2ℤ.