Abstract Alg, 28B: F[x] is a Principal Ideal Domain, Irreducibility over Integral Domains & Fields

Abstract Algebra, Lecture 28B.

(0:00) Math grad school preliminary written exams are usually extremely difficult.
(0:17) If F is a field, then F[x] is a principal ideal domain (PID).
(2:49) Outline of the proof that F[x] is a PID.
(8:02) Definition of an irreducible and reducible polynomial over an integral domain.
(12:32) Examples from the textbook.
(19:13) Degree 2 or 3 irreducibility test.
(20:00) If f(x) is a polynomial with integer coefficients that is irreducible over Z, then it will be irreducible over Q as well (and mention the Rational Root Theorem).
(22:09) Mod p irreducibility test and Eisenstein's criterion.

#ringtheory #irreducibility #principalidealdomain

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