Abstract Algebra Prelim Jan. 2010
1. Prove the rings Z/mnZand Z/mZ×Z/nZare isomorphic when mand nare relatively prime
(positive) integers. Discuss whether these rings are ever isomorphic when mand nare not
relatively prime.
2. Let S={(z, w)∈C×C:|z|2+|w|2= 1}. For a positive integer m, let Z/mZact on the set
Sby
(amod m)·(z, w) = e2π ia/mz, e8π ia/mw.
(a) Show this is a group action of Z/mZon S.
(b) If mis odd, show every orbit in this group action has melements.
(c) If mis even, show the orbit of some point in Shas less than melements.
3. Use Zorn’s lemma to show every nontrivial finitely generated group contains a maximal sub-
group. (A maximal subgroup is a proper subgroup contained in no other proper subgroup.)
Do not assume the group is abelian.
4. (a) Let abe any complex number. Prove that the map φ:R[x]→Cdefined by φ(f(x)) =
f(a) is a homomorphism of rings.
(b) Prove that R[x]/(x2+ 1) is a field which is isomorphic to C.
5. (a) Let Rbe a commutative ring with identity and Ibe an ideal in R. Show that R/I is a
field if and only if Iis a maximal ideal.
(b) Let Rbe a PID and Pbe a nonzero prime ideal in R. Show that Pis a maximal ideal.
6. Give examples as requested, with brief justification.
(a) A nonabelian group which is not isomorphic to a semidirect product of nontrivial groups.
(b) A 2-Sylow subgroup of S4.
(c) A PID other than Z.
(d) A unit other than ±1 in Z[√7].