{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Algebra Jan 1998 - Algebra Qualifying Examination January...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern