Abstract Algebra, Lec 27A: Facts about Ideals & Fields, Factor Ring Calculations (with Mathematica)

Abstract Algebra, Lecture 27A.

(0:00) Lecture plan.
(0:31) Basic facts to know about ideals, integral domains, prime ideals, and fields (ideals generated by elements, intersections of ideals, prime ideals in Z, ideals containing 1 or containing a unit equal the whole ring (they are not proper), prime ideals are maximal ideals in finite commutative rings with unity, if R is a commutative ring with unity then R is a field iff the only ideals are {0} and R, the ring of integers Z is a principal ideal domain (PID), in a PID every nontrivial prime ideal is a maximal ideal).
(17:57) x^2 + 1 is reducible over Z2 but irreducible over Z3 (it has a zero (root) in Z2 but it does not have a zero in Z3). If A is the principal ideal generated by x^2 + 1 (in each example), then Z3[x]/A is a field but Z2[x]/A is not a field (A is a maximal ideal in Z3[x] but not a maximal ideal in Z2[x]).
(21:13) Calculations in these factor (quotient) rings using simpler coset representatives. PolynomialMod in Mathematica is used.

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