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MATH 531
Some topics we covered in Chapters 12-16 AND
amended Homework
-
Rings (definition, elementary properties, unity=identity,
commutative rings, Cartesian product)
-
Units, Zero divisors
- Integral domains & fields
(definitions, fields are integral domains, finite integral domains are fields)
-
Examples of rings, integral domains, fields (many, including
,
,
,
,
,
,
,
,
,
) -
Subrings(definition, ways to verify, examples)
-
The ring R[i] where i2=-1; when this is a field
-
Characteristic of a ring; always prime for an integral domain
-
Finite fields have pn elements for some prime p, some
positive integer n; examples include
for certain
primes p
-
Polynomial rings, the basic operations, degree & its properties
-
The Division Algorithm for polynomials
-
The Remainder Theorem, the Factor Theorem, and the number of roots
of a polynomial (in an integral domain)
-
The Fundamental Theorem of Cyclic Groups (information about the
subgroups of a cyclic group)
Below is the amended homework for the ring theory material. (I
eliminated Chapters 17 and 18, and dropped a few problems in the
other sections.)
Chapter 12
p. 234
#3,5,6,7,8,9,14,15,19,20,22,23,25,26,29,30,34,35,36,38,44,45,47,48
Chapter 13
p. 246
#2,3,4,7,8,9,10,12,16,18,23,24,28,29,31a,34,36,37,41,49,54,55,
Supplementary Problems for Chapters 12-14
p. 267 #11,34,35
Chapter 16
p. 290 #2,3,11,12,17,20,31
Supplementary Problems for Chapters 15-18
p. 331 #13,35
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Allen D Bell
5/14/2004