Abstract Algebra Prelim Aug. 2010
1. Set Z[√11] = {a+b√11 : a, b ∈Z}.
(a) Show a+b√11 is a unit in Z[√11] if and only if a2−11b2=±1.
(b) Show there is no integral solution to a2−11b2=−1.
2. Let Gbe an abelian group and Hbe a subgroup with finite index. For any integer n≥1, the
inclusion H→Gand the reduction G→G/nG compose to give a homomorphism H→G/nG.
This homomorphism kills nH, so it induces a homomorphism fn:H/nH →G/nG given by
fn(hmod nH) = hmod nG.
(a) Whenever (n, [G:H]) = 1, show fnis an isomorphism.
(b) Whenever (n, [G:H]) >1, show fnis not surjective.
3. Let Snbe the symmetric group on nletters (n≥2). Then every element of Sncan be written
as a product of cycles.
(a) Write an arbitrary cycle (i1i2. . . im), m≥2, as a product of transpositions.
(b) Show that Snis generated by the n−1 transpositions (12), (13),...,(1n).
(c) Show that Snis generated by the n−1 transpositions (12), (23),...,(n−1n).
4. (a) In Q[x, y], prove that (x) is a prime ideal and not a maximal ideal.
(b) In Q[x, y], prove that (x, y) is a maximal ideal.
(c) In Q[x], prove that (x2−2) is a maximal ideal and that neither (x2) nor (x2−4) is a
prime ideal.
5. Let Abe a nonzero ring such that a2=afor all a∈A. (Examples include Z/2Z×·· ·×Z/2Z,
but these are not the only ones.)
(a) Show Ahas characteristic 2.
(b) If Ais finite, show its size is a power of 2.
(c) Show any prime ideal in Ais maximal.
6. Give examples as requested, with brief justification.
(a) Four nonisomorphic groups of order 8.
(b) A nontrivial character of the group (Z/12Z)×.
(c) A torsion-free Z-module which is not a free Z-module. (Torsion-free means no element v
satisifes nv = 0 for some nonzero integer nexcept for v= 0.)
(d) Three nonisomorphic C[x]-modules which are each 2-dimensional as C-vector spaces.