Algebra Ph.D. Qualifying Exam, January 2013
Answer all questions. Partial credit will be given.
1. Let pbe prime and let Pbe a group of prime power order pαfor some α≥1. Prove that the
center Z(P) is nontrivial.
2. Prove that x4+ 4x3+ 6x2+ 2x+ 1 is irreducible in Z[x].
3. Let pbe a prime. Compute the Galois group over Qfor the splitting field of the polynomial
xp−2, by describing the action of the elements of the group on generators of the extension.
4. Let Fbe a field and let VFbe a (possibly infinite-dimensional) vector space over F. Let
V∗= Hom(V, F ) denote the dual space of V. Consider the map
Φ : V→(V∗)∗, v 7→ Ev,
where Ev∈(V∗)∗= Hom(V∗, F ) is the evaluation map defined by the rule Ev(f) = f(v) (for
each f∈V∗). Prove that Φ is an injective linear transformation.
5. Show that the alternating group A5is simple.
6. Part A: Describe all ring homomorphisms (not necessarily sending 1 7→ 1) from Z→Q.
Part B: Describe all Z-module homomorphisms from Z→Q.
7. Prove the existence of a non-abelian group of order 20 not isomorphic to D20.
8. Let Ibe an indexing set and let Rbe a (commutative) ring with 1 6= 0. For each i∈Ilet Mi
be an R-module. Also let Nbe an R-module. Prove that there exists an isomorphism:
HomR M
i∈I
Mi, N !∼
=Y
i∈I
Hom(Mi, N ).
9. Let Rbe a noncommutative ring with 1. Assume ab = 1 but ba 6= 1. Prove that ahas at least
two distinct right inverses.
10. Let Fbe a field and assume α, β are both algebraic over F. Prove that α+βis algebraic over
F.
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