Abstract Algebra Prelim Jan. 2013
1. In the group Aff(Z/(7)) = {(a b
0 1 ) : a, b ∈Z/(7), a 6= 0}, compute the number of p-Sylow sub-
groups for each prime pdividing the order of the group.
2. Let Dn=hr, sibe the nth dihedral group for n≥3 (order 2n,rn= 1, s2= 1, sr =r−1s).
(a) For any automorphism fof Dn, show f(r) = rafor some integer asuch that (a, n) = 1,
and f(s) = rbsfor some integer b.
(b) Conversely, given integers aand bsuch that (a, n) = 1, show there is a unique automor-
phism fof Dnsuch that f(r) = raand f(s) = rbs.
3. Let Fbe an infinite field.
(a) If f(X)∈F[X] satisfies f(a) = 0 for all a∈F, then prove f(X) = 0 in F[X].
(b) If f(X, Y )∈F[X, Y ] satisfies f(a, b) = 0 for all (a,b)∈F×F, then prove f(X, Y ) = 0
in F[X, Y ].
4. (a) If a commutative ring Rhas exactly one maximal ideal, then prove this ideal must be
R−R×(the complement of the units in R).
(b) Let Rbe the ring of rational numbers with an odd denominator: R={a/b :a, b ∈
Z, b odd}. Describe R×and show Rhas a unique maximal ideal.
5. Let Abe a commutative ring with identity. For an ideal Iin Aand a∈Adefine
(I:a) = {c∈A:ca ⊂I}.
(a) Show (I:a) is an ideal in Aand it contains I.
(b) If the ideals I+Aa and (I:a) are both finitely generated then show Iis finitely generated.
More precisely, if I+Aa is generated by x1+b1a,. . . , xm+bma(xi∈I,bi∈A) and
(I:a) is generated by y1, . . . , yn, then show Iis generated by x1,. . . , xm, y1a, . . . , yna.
(c) Assume Acontains an ideal that is not finitely generated. Prove Acontains a prime ideal
that is not finitely generated. (Hint: Use Zorn’s lemma to show there is an ideal Pin A
that is not finitely generated and contained in no other ideal that is not finitely generated.
Then use part b to show Pis prime.)
6. Give examples as requested, with brief justification.
(a) A subgroup of Z×Zthat is not equal to aZ×bZfor integers aand b.
(b) A group isomorphism from Z/6Zto (Z/7Z)×.
(c) A ring isomorphism from R[x]/(x4−2) to R×R×C.
(d) A unit in Z[x]/(x3) other than ±1.