Ring Theory
Eisenstein's Criterion
A prime-divisibility condition on coefficients that certifies irreducibility of a polynomial over ℤ or any UFD.
Gauss's Lemma (Polynomial)
The product of primitive polynomials is primitive; irreducibility over a UFD transfers to irreducibility over its fraction field.
PID Implies UFD
Every principal ideal domain is a unique factorisation domain: elements factor uniquely into irreducibles up to units and order.
Euclidean Domain
A domain with a division algorithm: any a, b give a = qb + r with r smaller than b under a norm function.
Hilbert Basis Theorem
If R is Noetherian then so is R[x]: every ideal in a polynomial ring over a Noetherian ring is finitely generated.
Nakayama's Lemma
A finitely generated module annihilated modulo the Jacobson radical is zero — a powerful tool for lifting generators and splitting modules.
CRT for Rings
Coprime ideals split a quotient ring into a direct product: R/(I₁∩⋯∩Iₙ) ≅ R/I₁ × ⋯ × R/Iₙ.
Polynomial Division Algorithm
For any polynomial f and monic g, there exist unique q and r with f = qg + r and deg r < deg g.