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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
ringhomomorphismwithiterneldlndgnnniik. {a} Let 111,712,... .111. be positive integerssuch that [inmi} = 1 for all inf with i aij. Prove
that for any integersohom." .nt thereinsquZeuchthst
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 Rtnodule 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  sue1 — one? — s] has a root in every ﬁnite ﬁeld. E. [11] points] Let V == 111*, regarded as a2dimensional vector space over It. Let LW} denote
the ringof linear transformations from V to V. Let T : L(V} bedeﬂned byTIEr, y} = (1.33).
Let A={5'E Lﬂ")  ST=T.S'}. [i]: Prove that A is airing.
{ii} To what 1liIell known ring is A isomorphic? Give the isomorphism.
{iii} ChangeltoQandhenoel/aqa. Towhatringofnumhersisdisomorphie‘? ?.{15po_ints} LetFEE;LbeﬂeldssndsuppoeeorELisalgehrraicomF. Letﬂnl=
:“+ﬁnl:"‘l+u+ﬂre+ﬁnbetheminimal polynomialofomrer E. {a} Provethat iflyisarootoffﬁc} inanextension L'othhen‘riselgebraiLover 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 psubgroup 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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