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Algebra Jan 1998

# Algebra Jan 1998 - Algebra Qualifying Examination January...

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Unformatted text preview: Algebra Qualifying Examination January 1993 Directions: 1. Answer all questions. {Total possible is 101] points.) 2. Start each question on a new sheet of paper. 3. “Hits 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 exam may contain a. few misprints. If you are convinced a problem has been stated incorrectly. indicate your interpretation in 1Writing your answer. in such cases1 do not interpret the problem in such a way that it becomes trivial. Notes: AI] rings are unitary. All modules are unitary. Q is the rationals1 R the reels1 C the complexes, and Z the integers. l. [12 pts] Show that there are exactly two isomorphism classes of groups of order ‘21 and ﬁnd a presentation for both in terms of two generators [I of order 3 and y of order T}. 2. {15 pts} a} Determine the number of elements of each possible order in the icosahedral group 21; {also known as the alternating group on ﬁve letters). b} 1'Write the general form of the class equation which expresses the number of ele- ments in a ﬁnite group G as a certain sum that comes from the action of G on itself by conjugation. c] 1Write the class equation of A5, i.e. ﬁnd the numerical ‘I’ﬂnluES of the various terms. 3. [12 ptsj Let F27 be the ﬁnite ﬁeld with 27 elements. Vie-Wing FT.- as a vector space over F3. consider the map \‘F'i F21 —'* F27 I I-—-—r 5.3{1} = :3 a] Show that to is a vector space automorphism. b) Find a matrix representation for [P in G L3[F3}1 3 x 3 invertible matrices with entries in F3. 4. (12 pts} a) State the Fundamental Theorem of Galois Theory. bl Let fill = 5‘i + be: + c E F[:r] where F is a ﬁeld and let K be the splitting ﬁeld of F. Prove that the Galois group Giﬁ'jf‘} is contained in a dihedral group Dg. (You may assume the characteristic of F is not 2 if you wish.) 1 5. [12 pts} Let R he a commutative ring [with 13 E R] and let I E R he a proper ideal. Show that there exists a minimal prime ideal F over I. ie a prime ideal P such I E P and such that there does not exist another prime ideal P’ with IgP’EP. IE. [13 pts] Let R be a commutative ring [with 13 E R} and let I E R be a non-nilpotent element. Consider the multiplicative set 3 = {13.f1f21f3. . . .} and the localization 5 _1 R, also denoted H 1+ and the canonical homomorphism sz—rRI t1 ﬁll—“'— 1a a] Show there exists a natural l-l correspondence between prime ideals in R; and ' prime ideals in R which do not contain f. h] Give an example to show where this correspondence fails when we drop the “prime” assumption. T. [12 ptsj Let V be a complex vector space with a positive deﬁnite Hermitian form { , l: V x V —i C. This means1 among other things, that for 1:1 to E l" and c E C that {mun} = {1a. a} {cmw} = ﬂaw} {v1 cw} = Elﬁn. to} so { 1 l is C—linear in the ﬁrst factor and conjugate linear in the second factor. a] Let T: V —: V be a Herrnitian operator, Le. a {Cl—linear transformation satisfying {u,Tu:} = {ijw}. Prove that there exists a orthonormal basis of V consisting of eigenvectors. of T. b} Show that all the eigenvalues of the Hermitian operator T are real numbers. 3. {12 ptsj Let R he an integral domain. An R—module M is {limit-l: if for every m E M and every :- ii I] in R there exists an ﬁt in. M’ such that rnh = m. Recall also that an R- rno-dule I is injective if for every homomorphism tr: A —r I and every monomorphism ti: A —r B there exists a homomorphism ,3: B —r f such that the diagram ems-Ls ﬂ: 1 a I is commutative. Show that M injective implies M is divisible. 2 ...
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