Algebra Prelim, August 1997

Answer all questions

  1. Let R be a commutative ring with unity and let I, J be ideals of R.
    1. Prove that the product IJ = {x $ \in$ R | x = Si = 1naibi with ai $ \in$ I and bi $ \in$ J} is an ideal of R.

    2. Prove that IJ $ \subseteq$ I $ \cap$ J.

    3. If I + J = R, prove that IJ = I $ \cap$ J.

    4. If IJ = I $ \cap$ J for all ideals of R and R is an integral domain, prove that R is a field. (Hint: let I = Ra where a$ \ne$ 0 be a principal ideal or R.)

  2. Let F be a finite Galois extension of the field K with Gal(F/K) $ \cong$ S5.
    1. Show that there are more than 40 fields strictly between F and K.

    2. Show that there is a unique proper subfield E of F with E$ \ne$K such that E/K is a Galois extension. Determine [E : K] and describe Gal(E/K) up to isomorphism.

  3. Let G be a group of order 455.
    1. Prove that G is not simple.

    2. Prove that G is cyclic.

  4. Let R be a PID, let n be a positive integer, and let A and B be finitely generated R-modules. If An $ \cong$ Bn, prove that A $ \cong$ B. (An denotes the direct sum of n copies of A.)

  5. Let P be a finitely generated projective $ \mathbb {Z}$-module. If P is also injective, prove that P = 0.

  6. Let A, B be abelian groups, and let m be a positive integer. Prove that A $ \otimes$ (B/mB) $ \cong$ (A $ \otimes$ B)/m(A $ \otimes$ B).

  7. Prove that a group of order 588 is solvable.

  8. Let K = $ \mathbb {Q}$($ \sqrt{2}$ + $ \sqrt{3}$,$ \sqrt{5}$).
    1. Determine [K : $ \mathbb {Q}$].

    2. Compute Gal(K/$ \mathbb {Q}$).

Peter Linnell