Algebra Prelim, August 2015

Do all problems

1. Let G be the group of upper triangular invertible 3×3 matrices over the field 𝔽₅ of 5 elements.

   (a)

   Show that G has a unique Sylow 5-subgroup P₅.

   (b)

   Construct explicitly a composition series for P₅.

   (c)

   Show that G is isomorphic to the semidirect product P₂⋉P₅.

2. Let S be the ring of n×n-matrices with entries in a field F.

   (a)

   Show that the S-module V = 𝔽ⁿ of column vectors is a simple left S-module (simple means it is nonzero and has no submodules other than 0 and itself).

   (b)

   Show that every left ideal of S is a projective left S-module.

3. Find the Galois group of the polynomial f (x) = (x¹² -1)(x² - 2x + 2) over ℚ.

4. Classify the conjugacy classes of 5×5 matrices of order 3

   (a)

   with coefficients in ℚ.

   (b)

   with coefficients in ℂ.

5. Let M be a module over the integral domain R.

   (a)

   Prove directly that M = 0 if and only if M_P = 0 for all prime ideals P, where M_P is the localization of M at P.

   (b)

   Prove that an R-module homomorphism f : M→N is surjective if and only if, for all prime ideals P, the maps f_P : M_P→N_P are surjective, where by definition f_P(m/d )= f (m)/d for all m∈M and d∈R\P.

6. Let G be a group of order 2^ap, where 1≤a≤3 and p≥3 is a prime. Prove that G cannot be simple.

7. Let F = (F₁,..., F_m) be a system of m polynomial equations, where each F_i∈ℤ[x₁,..., x_r]. Consider the following statements:

   (a)

   The system F has solutions in ℤ.

   (b)

   The system F has solutions in ℤ/nℤ for any n≥1.

   (c)

   The system F has solutions in ℤ/p^sℤ where p is any prime and s≥1.

   (d)

   The system F has solutions in ℤ/pℤ for any prime p.

   Prove that each statement implies the next. Prove that (c) implies (b) and give counterexamples to the other five backward implications (d) implies (c), (d) implies (b), etc.