Polynomial Rings, Irreducibility and Unique Factorisation

Over a field F, divide f by a nonzero polynomial g by repeating four actions: divide the leading terms, add that term to the quotient, multiply g by it, then subtract. Stop when the remainder is zero or has smaller degree than g. The result f = qg + r is unique. Field coefficients let you divide by g's nonzero leading coefficient.

Division by x - a leaves a constant remainder. Substituting a into f = q(x)(x - a) + r gives r = f(a), the Remainder Theorem. Hence x - a is a factor exactly when f(a) = 0. A quadratic over a field is reducible exactly when it has a root there. Over the reals, a quadratic with no real root is irreducible; every real cubic has a real root and therefore a real linear factor.

For an integer-coefficient polynomial, a rational root u/v in lowest terms has u dividing the constant term and v dividing the leading coefficient. Test all such candidates. For degree 2 or 3 over Q, no rational root means irreducible; higher degrees can split without a linear factor. Irreducibility depends on the field: a rationally irreducible cubic still has a real root and factors over R.

An integer polynomial is primitive when its coefficients have gcd 1. For a positive-degree integer polynomial, Eisenstein's criterion applies if a prime p does not divide the leading coefficient, divides every other coefficient, and p^2 does not divide the constant coefficient. Then the polynomial is irreducible over Q. Missing coefficients are zero, which is divisible by p. Check all three divisibility conditions.

For primitive f in Z[x], reduce coefficients modulo a prime p. If the degree is preserved and the reduction is irreducible over F_p, then f is irreducible over Q. For reduced degree 2 or 3, test all field elements for roots. Higher degrees need other factor tests. For example, x^2 + x + 1 has value 1 at both 0 and 1 modulo 2, so its reduction is irreducible. A degree drop invalidates this test.

A principal ideal domain is a nonzero commutative domain in which every ideal has one generator. F[x] is one: in a nonzero ideal choose a nonzero g of least degree. Divide any f in the ideal by g. Its remainder also belongs to the ideal, so minimality forces it to be zero. Thus the ideal is (g); the zero ideal is (0). Repeated division also computes gcds, and the gcd generates the ideal (f, g).

In F[x], gcd division gives Bezout identities. If irreducible p does not divide a, gcd(p, a) = 1, so up + va = 1. If p divides ab, multiply this identity by b to see that p divides b. Thus irreducibles are prime. Repeated positive-degree splitting terminates, giving each nonzero polynomial a nonzero constant times irreducible factors. Primeness makes the factors unique up to order and nonzero constant multiples. Zero has no such factorization.