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Unformatted text preview: Algebra lQualifying Examination
January 1139? Directions: 1. Answer all questions. {Total possible is lﬂﬂ points.)
2. Start each question on a new sheet of paper. 3. Write only on one side of each sheet of paper.
Policy on Miaprints: The Qualifying Exam Committee tries to proofread the exams as carefully as pos—
sible. Nevertheless1 the exam may contain a few misprints. If you are convinced a
problem has been stated incorrectly. indicate your interpretation in writing your
answer. In such cases. do not interpret the problem in such a wayr that it becomes trivial. Notes: All rings are unitary. All modules are unitary. q is the rationals. R the reels. C
the complexes. and Z the integers. l. (Iﬂptsj Show there is no simple group of order 43. 2. [Iﬂpts] Determine the number of 3 x 3 invertible matrices over a ﬁnite ﬁeld having q
elements. 3. (lﬂpts) Let R be a commutative ring. Prove that all R—niodules are free if and only
if R is a ﬁeld. 4. (Eﬂptsj {a} Show that every ring R has a maximal ideal.
{bl Give an example of a ring with a unique maximal nonzero ideal.
{c} Give an example of a ring with a ﬁnite {is 1} number of maximal idea15.
{d} Give an example of a ring with an inﬁnite number of maximal ideals.
{e} Give an example of a noncommutative ring in which {ﬂ} is the unique maximal ideal. 5. {lﬁpts} Let K be the splitting ﬁeld of the polynomial f z: r3 — 2 E [MI].
a} Determine the degree [K : Q].
h} Determine the Galois group of the polynomial f.
c} Determine all subﬁelds of K. E. [lﬁpts] A Inmluie ELI over a ring R is said tn be uniform if for all I'lﬂﬂZlEl't'l suhmodules
ELC offlrf, BHC 79 {lit}.
:1} Give an example of a ring R and a. uniform Rrnodule M.
h) Prove that A is a uniform Rmoclule if and only if for an},r two nonzcro elements
lac E A there exist as E R such that I'll = 3c 315 II].
c] Prove that if A is a unifonn Rvmodule, and if U and V are simple Rsuhmodulcs
of :1, then I," = V. [Recall that a nonzero module is simple provided it contains no snhmoclules other than itself and the zero submodule. T. {lﬂpts} Let K be a ﬁeld of characteristic p and let ﬂat} = IF — r — c E er].
a} Show if n is a root of ﬁx), so is o + l.
b] Show that HM] is the splitting ﬁeld of ﬂat} 8. {lﬂpts} Let. G he a group of odd order. Prove that if n E G then there exists a unique
9 E G such that g} = :1. h.“ ...
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 Spring '08
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