Abstract Algebra Prelim Aug. 2011
1. (a) Prove the division algorithm in Z: if aand bare in Zand b6= 0 then there are qand rin
Zsuch that (i) a=bq +rand (ii) 0 ≤r < |b|. (In fact qand rare unique, but you don’t
need to show that.)
(b) Use part a to show every nonzero subgroup of Zhas the form nZfor a unique n≥1.
2. The commutator subgroup of a group G, denoted by G0, is the subgroup generated by all
commutators [x, y] = xyx−1y−1for all x, y ∈G.
Let p > 2 be an odd prime and define
G= a b
0c:a, c ∈(Z/pZ)×, b ∈Z/pZ⊂GL2(Z/pZ).
(a) Show that 1b
0 1 :b∈Z/pZis a cyclic group of order p.
(b) Show that G0is the group in part a.
(c) Show that G/G0∼
=(Z/pZ)××(Z/pZ)×.
3. (a) For a commutative ring Rand R-module M, define what it means to say Mis a cyclic
R-module.
(b) For any matrix A∈Mn(R), we can make Rninto an R[t]-module by declaring that for
any polynomial f(t) = c0+c1t+· ·· +cdtdin R[t] and vector vin Rn,f(t)v=f(A)v=
(c0I+c1A+· ·· +cdAd)v.
Determine, with explanation, whether Rnis a cyclic R[t]-module for each of the following
choices of A:
A=0−1
1 0 on R2, A =
230
020
002
on R3.
4. Show that a finite group whose only automorphism is the identity mapping must be trivial or
have order 2.
5. Let dbe a nonsquare integer and αbe nonzero in Z[√d] with norm N, so N=αα. Show
the principal ideal (α) in Z[√d] has index |N|. That is, show Z[√d]/(α) has order |N|. (Hint:
Consider the chain of ideals Z[√d]⊃(α)⊃(N).)
6. Give examples as requested, with brief justification.
(a) An infinite abelian group in which every element has finite order.
(b) An infinite field of characteristic p.
(c) An integral domain which does not have unique factorization.
(d) An irreducible polynomial in Z[t] of degree 8.