Polynomial irreducibility is never just about the polynomial—it depends on the coefficient ring or field. Learn a complete Abstract Algebra decision system using root tests, the Rational Root Theorem, Gauss’s Lemma, Eisenstein’s Criterion, reduction modulo p, finite-field factor tests, quotient rings, and field extensions.
Field extensions are how Abstract Algebra builds a bigger number system when a polynomial has missing roots. In this Woody Calculus visual lesson, learn how Q becomes Q(√2), why x² − 2 has no rational root, how degree and minimal polynomials work, and why field extensions open the door to Galois theory.
