MASTERS EXAM (SPRING 2002)
ALGEBRA
Answer any three of the following five questions. You must state clearly any general
results you use.
1. Which of the following pairs of groups are isomorphic. Give reasons.
(a) (R,+) and (R+,×) where R+={x∈R:x > 0},
(b) (Q,+) and (Q+,×) where Q+={x∈Q:x > 0},
(c) (C×,×) and (C/Z,+) where C×={x∈C:x6= 0}.
2. Let Gbe a finite group.
(a) Show that Gis not the union of two proper subgroups.
(b) Show that Gis the union of three proper subgroups iff there exists a surjective
homomorphism from Gto Z/2Z×Z/2Z. [Hint: If the subgroups are H1,H2,
H3, show that H1∩H2⊆H3.]
3. Prove that any group of order 15 is cyclic.
4. Let dbe a squarefree integer and R=Z[√d].
(a) Show that if Iis a non-zero ideal of Rthen there exists an integer n6= 0 with
n∈I.
(b) Deduce from (a) that if Iis a non-zero ideal of Rthen R/I is finite.
(c) Deduce from (b) that any non-zero prime ideal of Ris maximal.
5. Let Fbe a field and let R=Mn(F) be the ring of n×nmatrices with entries in F.
Show that Rhas no non-trivial proper ideals.