{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Math 113 - Fall 2002 - Bergman - Final

A First Course in Abstract Algebra, 7th Edition

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: 05/12/2003 MON 16:26 FAX 6434330 MOFFITT LIBRARY .001 George M. Bergman Fall 2002, Math 113, Sec. 5 13 Dec., 2002 9 Evans Hall Final Examination 12:30—13:30 1. (12 points, 4 points each.) Complete the following definitions. (a) If X is a set, then a permutation of X means (b) If R and S are commutative rings, then a map f : R —) S is called a homomorphism of commutative rings if (c) A proper ideal I of a commutative ring R is said to be a maximal ideal if 2. (36 points; 4 points each.) For each of the items listed below, either give an example, or give a brief reason why no example exists. (If you give an example, you do not have to prove that it has the property stated. Examples should be specific for full credit; i.e., even if there are many objects of a given sort, you should name one.) (a) An element o-E S6 such that 0' (1 2 3) 0‘1 = (2 4 6). (b) An injective (i.e., one-to—one) homomorphism f : Z —> lRX. (Recall that le denotes the group of nonzero real numbers under multiplication.) (c) A factorization of the polynomial 3 x3 + 29 x2 — 4x — 2 as a product of two polynomials of lower degree in ©[x]. (d) A ring R, an ideal I g R, and elements a at b of R such that a+I = b+I. (e) A polynomial f(x) E(D[x] which has no root in (D, but which is reducible in ©[x]. (f) A field with exactly 100 elements. (g) An ideal I g Z[x] such that Z[x]/I is isomorphic to Z[i], the ring of Gaussian integers. (h) Two elements of ZS [x] that are associates, but are not equal. (i) A unique factorization domain which is not a principal ideal domain. 3. Short proofs. (22 points : 6+8+ 8.) (a) If f: X —> Y and g : Y —> Z are set maps such that the composite map 30f: X—> Z is injective, Show that f is injective. (b) Suppose a group G acts on a set X, and let S = {06 G | V x6 X, ox = x}. Show that S is a normal subgroup of G. (You must show both that it is a subgroup and that it is normal.) (0) Suppose P1 2 P2 2 2 P” 2 PH” 2 is a decreasing sequence of prime ideals. Show that the ideal I = final Pn is prime. (Here you are to take for granted that I is an ideal, in contrast to the homework problem this is taken from, where you had to prove both that it was an ideal and that it was prime.) ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online