Algebra Prelim, May 2006

Do all problems

1. Prove that there are no simple groups of order 1755.

2. Let P be a finite p-group. Prove that every subgroup of P appears in some composition series for P.

3. Let R be a principal ideal domain. Let A be a finitely generated R-module and let B be an R-submodule of A. Assume that there exist nonzero elements r and s of R such that gcd(r, s) = 1, rB = 0, and s(A/B) is a torsion-free R-module. Prove that A @ B $\oplus$ A/B as R-modules.

4. Let F = Q(i) and let K be the splitting field of x⁶ - 7 over F.

   (a)

   Determine [K : F] and write down a basis for K over F.

   (b)

   Show that Gal(K/F) is a dihedral group.

5. Let R be a ring with unity 1. Let P be a projective R-module and let M be an R-submodule of P. Prove: if P/M is a projective R-module, then M is a projective R-module.

6. Let R be a commutative Noetherian ring with unity 1. Let M be a nonzero R-module. Given m e M, set Annm = {a e R | am = 0} and note that Annm is an ideal of R. Prove that there exists s e M such that Anns is a prime ideal in R. (Remember: R itself is not a prime ideal in R.)

7. Let A = C $\otimes_{{\mathbf{R}}}^{}$ C.

   (a)

   Prove that there is a well-defined multiplication on A that satisfies the distributive property such that

   (a₁ $$\otimes$$ b₁)(a₂ $$\otimes$$ b₂) = a₁a₂ $$\otimes$$ b₁b₂

   for all complex numbers a₁, a₂, b₁, b₂.

   (b)

   Now assume that this multiplication makes A into a ring. Prove that A is not an integral domain.