Final Exam, Math 200, Fall 2006
(Robert Boltje)
1. (a) Define the term “Euclidean Domain”.
(b) Determine all units of Z[i].
(c) Show that Z[i] is a Euclidean domain.
(d) Compute a greatest common divisor dof a:= 11 + 3iand b:= 7 −2i
in Z[i] and express it as a Z[i]-linear combination of aand b.
(e) Determine if p:= 7 + 2iis a prime element or an irreducible element
in Z[i].
2. (a) Define what it means for a group Gto be “nilpotent”.
(b) Show that an abelian group is nilpotent.
(c) Is Sym(3) nilpotent?
(d) Show that every nilpotent group is solvable.
(e) Show that if Gis a p-group and Na normal subgroup of G, then
Z(G)∩Nis not trivial.
3. (a) Show that X4+ 1 is irreducible in Q[X].
(b) Show that R:= Q[X]/(X4+X2) is isomorphic to the direct product
of two rings.
(c) Compute two non-zero elements e,f ∈Rwith the property e2=
e, f 2=f, ef = 0, e +f= 1.
(d) Find all maximal ideals of R.
4. (a) Let Rbe a commutative ring and let Sbe a multiplicatively closed
subset of Rthat contains 1. Define ι:R→S−1R,r7→ r
1. Show that ιis
injective if and only if Sdoes not contain any zero-divisor.
(b) Prove or find a counterexample to the following statement: If f:R→
Sis a unitary ring homomorphism and Mis a maximal ideal of S, then
f−1(M) is a maximal ideal of R.
(c) Let Rbe an integral domain and let a, b ∈R. Show that (a) = (b) if
and only if aand bare associate.
(d) Let R:= Z[√5i] and let Ibe the ideal of Rgenerated by 3 and
1 + √5i. Show that Iis a maximal ideal.
5. (a) Let Gbe a group such that G/Z(G) is cyclic. Show that Gis abelian.
(b) Show that every group of order p2is abelian.
(c) Let p<qbe primes such that pdoes not divide q−1. Show that
every group of order pq is cyclic.
(d) For a prime pand a finite group Glet Op(G) denote the intersection
of all the Sylow p-subgroups of G. Show that Op(G) is normal in G.
Each of the 5 problems is worth 10 points.
