Modern Algebra II MA 542 Spring 2010
Meetings: GCB 206 (750 Comm Ave), MWF 1:00pm2:00pm Instructor: Robert Harron Email rharron@ Oce MCS 230 Oce Hours M 2:30pm3:30pm F 3:00pm4:00pm Course website: http:/math.bu.edu/people/rharron/MA542/ Textbooks: Contemp
Assignment 12 MA 542 Due in class: Wednesday, Apr. 28, 2010
(1) (a) If F has characteristic 0, show that Aut(F ) = AutQ (F ), i.e. show that every automorphism of F xes every rational number. (b) If F has characteristic p, show that Aut(F ) = AutFp (F ).
Assignment 11 MA 542 Due in class: Thursday, Apr. 22, 2010
Chapter 21: 4
(Edition 6: same number) (Edition 6: 2, 8, 10, 24, 30)
Chapter 22: 2, 12, 14, 28, 34
(1) Let F, K, and L be elds with F K L. If L is a nite extension of F and [L : F ] = [L : K ], sh
Assignment 10 MA 542 Due in class: Wednesday, Apr. 14, 2010
Chapter 21: 12, 14, 16, 22, 32, 34, 35
(Edition 6: same numbers)
(1) Find the minimal polynomial of over F in the following situations: (a) = e2i/3 , F = Q (b) = i 2, F = Q( 2) (c) = i 2, F = Q (
Assignment 9 MA 542
Due in class: Wednesday, Apr. 7, 2010
Chapter 20: 30, 32, 34
(Edition 6: same numbers)
35) Let F, K , and L be elds with F K L. If L is a splitting eld for some
nonconstant polynomial f (x) over F , show that L is a splitting eld for f
Assignment 8 MA 542
Due in class: Wednesday, Mar. 31, 2010
Chapter 20: 2, 7, 8, 10, 16, 30, 32
(Edition 6: same numbers)
(1) Let R be a commutative ring with identity. Suppose R is not an integral domain.
Give an example of a polynomial over R that doesnt
Assignment 7 MA 542
Due in class: Wednesday, Mar. 24, 2010
Chapter 19: 9, 10, 22, 24
Chapter 20: 20
(Edition 6: same numbers)
(Edition 6: same numbers)
(1) Let U and V be two vector spaces over a eld F . Let T : U V be a linear
transformation.
(a) Show th
Assignment 6 MA 542
Due in class: Friday, Feb. 26, 2010
Chapter 18: 8, 14, 15, 18, 22, 32, 33, 36
(Edition 6: same numbers)
(1) Determine whether every non-zero prime ideal in a PID is maximal.
(2) Let R = Z[i] and d be its size function d(x + yi) = x2 +
Assignment 5 MA 542
Due in class: Wednesday, Feb. 17, 2010
Chapter 17: 2, 10, 12, 18, 30
(Edition 6: same numbers)
(1) Figure out all irreducible polynomials of degree 5 in (Z/2Z)[x].
Assignment 4 MA 542
Due in class: Wednesday, Feb. 10, 2010
Chapter 14: 6, 62
(Edition 6: 6, 58)
Chapter 16: 2, 4, 12, 40, 42
(Edition 6: 2, 4, 12, 38, 40)
(1) Show that I = cfw_(5x, y ) : x, y Z is a maximal ideal of Z Z.
(2) Give an example of two prime
Assignment 3 MA 542
Due in class: Wednesday, Feb. 3, 2010
Chapter 13: 58 (For edition 6, this is number 54)
15.46) (Chapter 15, # 46) Show that a homomorphism from a eld onto a ring with
more than one element must be an isomorphism. (Note that the homomor
Assignment 2 MA 542
Due in class: Wednesday, Jan. 27, 2010
Chapter 13: 6
Chapter 14: 4, 18, 29, 39, 45, 46
(1) Let R = R[x] and let I = (x2 ). Find a nilpotent element in R/I .
(2) Let R be a commutative ring with identity. An idempotent in R is an elemen
Assignment 1 MA 542
Due in class: Wednesday, Jan. 20, 2010
Chapter 12: 8, 20, 23, 24, 28
(1) Let R be a commutative ring with unity. Show that a R is a unit if, and only if,
there exists b R such that ab is a unit.
(2) (a) An element x of a ring R is call