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Unformatted text preview: Algebra Qualifying Exam, January 1999 Directions:
{1] Answer all questions. ['Ibtal possible is till} points.)
{2] Start each question on a new sheet of paper. {3} Write only on one side of each sheet of paper. Notes: All rings and motiules are unitary. The rational numbers vviii be Lienoteci by Q1 the real
numbers by It1 the complex numbers by C, the integers by Z, the non~negative integers {including [i] by N, the positive integers by N”, and the ﬁnite ﬁeicl with p elements by FF. 1+ {ltl points}
{i} vae that a group with only a linite number of subgroups must be ﬁnite.
{ii} Prove that a nontrivial group is inﬁnite cyclic it and only if it is isomorphic to each of its nontrivial subgroups. 2+ {15 points} {1} List all the nontrivial Sylow subgroups of 55, the set of all permutations on five letters. The
subgroups must be given expiicitly n in terms of a generating set for example. {ii} Is 5'5 nilpotent T Prme or disprove. {iii} Is 3!. solvable ‘5' Prove or disprove. 3. (5 points]
Let R be a commutative integral domain containing a ﬁeld i: as a snbring. Suppose that R is Einite dimensional when viewci as a investor space. Prove that R is a ﬂeltl. 4. {it} points] {a} Let I be an ideal in the polynomial ring 3 = ﬂinch. Nara]. The radical of I in S is the icleal
1.2“? := {f E S : f“ E I for some in E {Ti}. Prove that we“? is the intersection of all prime ideals in
3 that contain I. {b} A monoinial in S is a polynomial in S with only one term and unit ceeﬂicient. For example is a uronomial in Hahn]. A IIunrnmial in 5' is usually cienobed an as“ where o: E N“ and :s" = ril‘agi a mag“. A. mononiiai ideai in S is an ideal in S that is generated by monomiais. Let M = {Iﬂ1l...l.rmli be a moaemioi‘ idea! in 5'.
{il Find a generating set for vM in terms of the generating set for M.
{ii} Is the radical of Mr again a monomial ideal in S 'i' If not. provide a counterexample. {iii} If M is a prime inonemial idesJ. then what is the general form of M '? 5. {5 points} Let R be a ring {with 1}. Let M be a simple {unitalj Ftrnodule. Prove that there exists a rnaxitnal left ideal L of R such that Rf L E M as Fitmodules. I5. [11] points} Let R be a ring, and let P be an R—module. Shim that the following are equivalent:
{i} P is projeetive, [ii] Ever]; short Exact sequence [1 —} A —} B —} P —i‘ [I is split exact. {111] There exists a tree R—Inodule F and an ernodule if such that F’ E H E) P. 7. {ID points]
Let it” he. the splitting ﬁeld over IQ of the polynomial r“  2:2 — 1. Determine the lGalois group G of lift}; ﬁnd all intermediate ﬁelds and match them up with the subgroups of G. 3. {25 points}
Let p be a prime and let a be a rational number that is not a pth power. Let K be the splitting
ﬁeEd of the polynomial .rl’  a over Q. [i] Prove K is generated by apth root o of a and a primitive pth root .5 of unity. {ii} Prove [K : C1] = p{p— 1}.
{iii} Prove that the Galois group of Kit} is isomorphic to the group of invertible 2 x 2 matrices of n i: a I]
the form . with entries in FF. Explicitly describe the action of the elements and 01 lEll 1 El
on. the generators. II] 1
{iv} If p = '? how many intermediate extensions are there of degree equal to {i ever Q ’5' ...
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