Separability: count distinct conjugates, not repeated copies

The formal derivative applies the rule that the derivative of Xⁿ is nX raised to n − 1, with coefficients interpreted in the field. No limit or metric is required. If α is a repeated root, write the polynomial with a factor (X − α)² and differentiate: both the polynomial and its derivative vanish at α. Conversely, if the polynomial and derivative share a root in a splitting field, the corresponding linear factor occurs with multiplicity at least two.

Consequently a nonzero polynomial is separable, meaning all its roots are distinct, exactly when its greatest common divisor with its derivative is one, normalized to be monic. This criterion can be checked over the original field using the Euclidean algorithm; explicit roots are unnecessary. For an irreducible polynomial, a nontrivial common divisor must be the entire polynomial. Since a nonzero derivative has smaller degree, an irreducible polynomial is inseparable precisely when its derivative is zero.

In characteristic p, the coefficient p equals zero. Thus the derivative of Xᵖ − t is zero, even though the polynomial is nonconstant. Take F = Fₚ(t), the rational function field in a transcendental indeterminate t. The polynomial Xᵖ − t is irreducible: Eisenstein at t applies in Fₚ[t][X], followed by Gauss's lemma to pass to its fraction field. A root α therefore has degree p over F.

Nevertheless the binomial identity in characteristic p gives Xᵖ − t = (X − α)ᵖ in the field containing α. There is just one distinct root. Every F-embedding must send α to a root of its minimal polynomial, so there is only one such embedding into an algebraic closure. This is a purely inseparable extension. It demonstrates that degree counts linear coordinates even when repeated roots prevent those coordinates from corresponding to different conjugate choices.

An algebraic element is separable over F when its minimal polynomial over F has distinct roots. A field extension is separable when every element is separable. A polynomial with repeated factors can be inseparable as a polynomial even when all of its individual roots are separable elements. For instance, (X − 1)² over Q has a repeated root, but the minimal polynomial of its root 1 is X − 1, which is separable. Never use a redundant equation to diagnose an element.

Every irreducible polynomial in characteristic zero has nonzero derivative and hence is separable. Finite fields also have only separable irreducible polynomials: their Frobenius map a ↦ aᵖ is injective, therefore surjective on the finite set. If an irreducible nonconstant polynomial had zero derivative, every exponent would be a multiple of p and every coefficient would be a pth power. The polynomial would itself be a pth power, contradicting irreducibility. Thus positive characteristic permits inseparability without forcing it.