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Algebra Jan 1997

# Algebra Jan 1997 - Algebra lQualifying Examination January...

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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 idea-15. {d} Give an example of a ring with an inﬁnite number of maximal ideals. {e} Give an example of a non-commutative 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'lﬂﬂZl-El't'l suhmodules ELC offlrf, BHC 79 {lit}. :1} Give an example of a ring R and a. uniform R-rnodule M. h) Prove that A is a uniform R-moclule 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 R-suhmodulcs of :1, then I," = V. [Recall that a nonzero module is simple provided it contains no snhmoclules other than itself and the zero submo-dule. 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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Algebra Jan 1997 - Algebra lQualifying Examination January...

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