Algebra Prelim, August 2019

Do all problems

1. Prove that there is no simple group of order 4860 = 4·3⁵·5.

2. Prove that 2x³ +19x² - 54x + 3 is irreducible in ℚ[x].

3. Let R be a subring of the (not necessarily commutative) nonzero ring S. Assume that R and S have the same 1. Prove that S⊗_RS≠ 0.

4. Let R be a PID which is not a field, and let I be a finitely generated injective R-module. Prove that I = 0.

5. Determine the number of conjugacy classes of matrices in GL₈(ℚ) that consist of elements of order 7.

6. Let f∈ℚ[x] be a polynomial over ℚ, let K be a subfield of ℂ which is a splitting field for f over ℚ, let p be an odd prime, and let γ : ℂ→ℂ denote complex conjugation. Assume that deg f = 5, Gal(K/ℚ)≌S₅, and that e^2πi/p∈K.

   (a)

   Prove that f is irreducible and has exactly 5 distinct roots.

   (b)

   Prove that γ permutes the roots of f and that p = 3.

   (c)

   How many roots of f does γ fix? Prove that f has exactly two complex roots and three real roots.

7. Let H be a normal subgroup of index 3 in the finite group G, let x∈G∖H, and let χ be an irreducible complex character of H. Suppose that χ(xhx^-1) = χ(h) for all h∈H. Prove that Ind_H^Gχ (the induced character) is the sum of 3 distinct irreducible characters of G.