Algebra Prelim, Spring 1990

Answer all questions

1. (a)

   Let M be a left R-module, and let A and B be Artinian submodules. Show that A + B is an Artinian R-submodule.

   (b)

   If R is also left Noetherian and M is finitely generated, show that M has a unique maximum Artinian submodule A(M) and that A(M/A(M)) = 0.

2. Let A be an abelian group with no elements of infinite order. Suppose that every element of prime order is of order 3. Show that the order of every element is a power of 3. (Hint: do finitely generated abelian groups first.)

3. Let G be a simple group of order 144.

   (a)

   Prove that a group of order 18 has exactly one Sylow 3-subgroup.

   (b)

   If H is a proper subgroup of G, show that | H|≤26.

   (c)

   If P and Q are distinct Sylow 3-subgroups of G, show that | P∩Q| = 1. (If | P∩Q| > 1, consider N_G(P∩Q)).

4. Prove that a group of order 765 is abelian.

5. Let f (x) in Q[x] be an irreducible polynomial of degree 5. Suppose a and b are distinct roots and that Q(a) = Q(b). Show that Q(a) is a normal extension of Q.

6. Let S⊆Z[x₁, x₂,..., x_n]. Prove that there is a smallest principal ideal containing S. If this ideal is generated by α, show that αQ[x₁, x₂,..., x_n] is the smallest principal ideal in Q[x₁, x₂,..., x_n] containing S.

7. Let V be a vector space over R, and let T : V -> V be a linear transformation. Describe how V can be made into an R[x]-module.

   Now suppose there are v₁,..., v_n in V such that {Tⁱ(v_j) | i = 0, 1,..., j = 1, 2,..., n} span V as a vector space.

   (a)

   Prove that V is a finitely generated R[x]-module.

   (b)

   If T is onto, show that V cannot have a summand isomorphic to R[x].

   (c)

   If T is onto, show that V is finite dimensional.

8. (a)

   Let A = {ι,(1 2)(3 4)} and V = {ι,(1 2)(3 4),(1 3)(2 4),(1 4)(2 3)} in S₄. Show that A is normal in V and V is normal in S₄, but A is not normal in S₄.

   (b)

   Give an example of fields F⊆K⊆L such that K a normal extension of F and L a normal extension of K, but L is not a normal extension of F.