Algebra Ph.D. Qualifying Exam, August 2012
Answer all questions. Partial credit will be given.
1. Prove that there is no simple group of order 105.
2. Let Rbe a Euclidean domain. Prove that every nonzero prime ideal contains a prime element.
3. Find a primitive element for the extension Q(3
√2, ω)/Q, where ωis a primitive cube-root of one.
(You must prove that the element you found really is a simple generator for the extension.)
4. Find the characteristic polynomial, the minimal polynomial, and the invariant factors (over Q)
of the matrix
110
111
011
. Use these to find the Jordan canonical and rational canonical forms.
5. Let pbe an odd prime. Describe four non-isomorphic groups of order p3, and prove they are
non-isomorphic.
6. Let Rand Sbe commutative rings with 1 6= 0. Prove that the ideals of R×Sare precisely the
sets of the form I×Jwhere Iis an ideal of Rand Jis an ideal of S. In particular, prove that
R×Sis never a field.
7. Prove that the center of a nontrivial, finite p-group is always nontrivial.
8. Prove that an infinite dimensional vector space Vover F2is not isomorphic to its dual V∗=
HomF2(V, F2). (You may freely use true facts about cardinalities, if it is helpful.)
9. Let Rbe a commutative ring with 1 6= 0. Prove that if f(x)g(x) = 0 for some nonzero polyno-
mials f(x), g(x)∈R[x] then there exists a nonzero element c∈Rsuch that f(x)c= 0.
10. Prove that irreducible polynomials over Qare always separable. Do the same for polynomials
over finite fields.
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