MTH 1322 Rings, Fields, and Number Theory
Classnotes - Prof. A. Iarrobino
Meeting Mon,Wed, Thurs 10:30 AM in 509 Lake.
Look here for specific assignments and brief comments on what we did (or will do) in class. This will be updated at least
weekly. TBA = "to be announced". Note that you are not expected to do all assigned homework, but are welcome to try all.
I will collect HW once a week.
Wednesday March 27, Thurs. March 28. (review of groups):
Structure of groups of small orders. Order of subgroup divides order of group. Reason why groups of prime order are
cyclic. Normal subgroups: example A_3 (even permutations, so rotations and e for S_3) is normal in S_3, but
the order two subgroups are not normal. Examples of groups in low order. Classification of groups of
order 14, or 2p (begun).
HW: Worksheet #1. Also text Chap 10, continuation of 1321 HW.
Monday April 1: Intro to rings and fields. Definition of ring. Criterion for subset of a ring to be a subring.
Examples of noncommutative rings: group ring, matrix ring.
Examples of commutative ring without 1: 2Z. Examples of ring with nilpotents: k[x]/(x^2) or k[x]/x^3.
The ring F= Q+Q(sq rt(2)) isomorphic to Q[x]/(x^2-2) (this is an example of constructing field extensions).
The inverse of a nonzero element a+b(sq rt(2) in F is (a-b(sq rt(2))/(a^2-2b^2), so is in F. Description of
the automorphism group of Aut_Q(F) = (e,\sigma), with \sigma(sq rt (2)) = -sq rt(2).
HW for Thursday April 4: p. 234 in Gallian 5th edition: #1,2,3,4,6,7,13,15,17,18,20,28.
Also, describe Q(cube rt 3) (basis over Q) and show that Aut_Q(Q (cube rt 3)=(e). Also describe
Q((complex cube rt (1)) (basis over Q) and show that Aut F/Q is Z_2.
General info, grading, etc
See Math 1322 Information
Prof. Anthony Iarrobino
Department of Mathematics
Northeastern University
Boston, MA,
02115
Office:526 NI
Phone: (617) 373-5524
Email: iarrobin@neu.edu
Created: March 28,2002. Last modified: March 28,2002.
URL:http://www.math.neu.edu/~iarrobino/AIMath1302.Spr02