Algebra Ph.D. Qualifying Exam, January 2012
Answer all questions. Partial credit will be given.
1. Prove that a free group on any generating set with more than one element is non-Abelian.
2. Prove that every commutative ring with 1 6= 0 has a minimal prime ideal. You may freely use
the fact that such rings have maximal ideals if that is helpful.
3. Compute the Galois group over Qfor the splitting field of the polynomial x4+x3+x2+x+ 1.
4. Find the rational canonical form and Jordan canonical form for the integer matrix
123
456
789
.
5. Prove that Aut(Q8)∼
=S4where Q8=hi, j, k :i2=j2=k2=−1,(−1)2= 1, ij =k=
−ji, ki =j=−ik, jk =i=−kjiis the quaternion group. (You are free to use any presenta-
tion of this group you like.)
6. Prove that a PID has the property that any ascending chain of ideals stabilizes.
7. Classify all groups of order p2, where pis prime.
8. Let Fbe a field. Let Vbe a finite dimensional vector space. Prove that Vis isomorphic to its
double dual (V∗)∗.
9. Is the (possibly infinite) direct product of fields a PID? Justify your answer.
10. Let pbe a prime. Prove that there is a field of order pn, for each n≥1.
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