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Algebra Jan 2000

# Algebra Jan 2000 - Algebra Qualifying Examination January...

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Unformatted text preview: Algebra Qualifying Examination January, 21100 Directions: 1. Answer all questions. {Total possible is 10!] points.) 2. Start each outstion on s new sheet of paper. 3. Write only on one side of each sheet of paper. Policy on Misprints: The Qualifying Exam Committee tries to proofread the exams as carefully as possible. Nevertheless, the em may contain a few misprints. If you are oonirineed a problem has been stated incorrectly, indicate your interpretation in writing your answer. In such cases. do not interpret the problem in such a way that it becomes trivial. Notes: 1. All rings are unitary. All modules are unitary. 2. Q is the rationals, R the reels, C the titumliltttee1 and Z the integers. 1. [It] points} Let G be ngroup. Let H and N besuhgroups ofG with N normal in G. Prove that HHN n H} a How. 2. [2t] points] Recall theChinme Remainder Theorem: Let Rheaoommutstive ringwith 1. Let A1.A1.....At heideala ofRsuch that ifigﬁjthen A5+Aj =R. The: themsp RﬂRfAIQRfA2\$-'r\$ﬂfﬂh deﬁnedbyr—v[r+A1,r+A1,...,1-+A1:]isasurjective ringhomomorphismwithiterneldlndgnn-niik. {a} Let 111,712,... .111. be positive integerssuch that [in-mi} = 1 for all inf with i aij. Prove that for any integersohom." .nt thereinsquZ-euchthst zanlmodm zangmdtta a: a at mod nt. _ (ii) Let in. and n he poeitive integers. Prove that Ifmz —- anz deﬁned by c+m1 - a+nZ issuing homomorphismii'sndoulyii'ndividesm. {c} Let m and n be positive integers with n dividing; m. Prove that the natural surjective ring homomorphism mez —+ ﬂu: takes the group of units (meZJ' of mez onto the group of units (Bird? of anz. 3. {15 points} Let K and F be ﬁelds 1with It being; a ﬁnite extension of F. {a} Deﬁne the Galois group, Gal{K ,1“ FL of K over P. {hi In general, how are the numbers lellIKfFH and [K : F] related? [o] Deﬁne what it means for K to be a Galois extension of F. Now assume that K is a Galois extension of F. Let 33: be the set of all suhﬁelds of K containing F and let 3.»; he the set of all subgroupsof G. Aeoording to the Fundamental Theorem of IGsdois Theoryr there is a bijeotion 1': SK —- 3.9. {d} Deﬁne 9, Le. deﬁne ME} where E E SE. {e} What is triﬂe? where H E 3.3? {f} If E E SE when is K a Galois extension of E? (g) IfEESx whenis EaGabis extension ofF? [11} If X is Galois on: E idlenti:l'jr the group GnIIIIKIKE]. {i} If E is Galois over F identify the group Gsl{E,:‘F). {j} True or false: HKsnd Fare ﬁnite ﬁelds with K 2 F. then If is Galois over F. Justify,r your answer. 4. [ll] points] Let R be an integral domain. Let M be a torsion Rvmodule. {i} Prove that if M is ﬁnitelyr generated then there exists a nonzero r E R with rm = I] for everymEM. _ {ii} Give an example of a torsion R-tnodule M such that if E R with ma = U for every mEMthenr=D. 5. [It] points} {i} Prove that at least one of 2. 3 or ii is a square in the ﬁnite ﬁeld GFIEp} for every prime p. {ill Conclude that [:1 - sue-1 — one? — s] has a root in every ﬁnite ﬁeld. E. [11] points] Let V == 111*, regarded as a2-dimensional vector space over It. Let LW} denote the ringof linear transformations from V to V. Let T :- L(V} bedeﬂned byTIEr, y} = (1.3-3). Let A={5'E Lﬂ") | ST=T.S'}. [i]: Prove that A is airing. {ii} To what 1li-Iell known ring is A isomorphic? Give the isomorphism. {iii} ChangeltoQandhenoel/aqa. Towhatringofnumhersisdisomorphie‘? ?.{15po_ints} LetFEE;LbeﬂeldssndsuppoeeorELisalgehrraicomF. Letﬂnl= :“+ﬁn-l:"‘l+-u+ﬂre+ﬁnbetheminimal polynomialofomrer E. {a} Provethat iflyisarootoffﬁc} inanextension L'othhen‘riselgebraiL-over F. (b) Prove that all the ooeﬁ'icients of ﬂat] are algebraic over F. E. If It] points} Let G be a. ﬁnite group with Sylow p-subgroup P. Let Mali?) he the uormsliser of F in G. Prove that ifH is asuhgroup of»? that oontslns ﬁnk-{P}. then H equals its own normaliser. ...
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