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.
    Determine [K : F] and write down a basis for K over F.
    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.
    Prove that there is a well-defined multiplication on A that satisfies the distributive property such that

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

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

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

Peter Linnell 2006-05-22