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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 $\mathbb {Z}$ , $\mathbb {Q}$ , $\mathbb {R}$ , $\mathbb {C}$ , $\mathbb {Z} _n$ , $\mathbb {Z} [i]$ , $2 \mathbb {Z}$ , $M_2( \mathbb {Z} )$ , $\mathbb {Q} [\sqrt{d}]$ , $R\oplus S$ )
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 $\mathbb {Z} _p[i]$ 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)