15.1. Introduction🔗

HexGFqField constructs the field GF(pⁿ) = Fₚ[x] / (f) for a prime p and an irreducible degree-n modulus f of type Hex.FpPoly. It builds on the quotient ring documented in the HexGFqRing chapter: the field element type Hex.GFqField.FiniteField wraps a single Hex.GFqRing.PolyQuotient value, and every operation delegates to quotient-ring arithmetic and re-reduces, so the canonical-representative invariant from HexGFqRing carries over unchanged.

What the field adds over the ring is the structure that only exists when the modulus is irreducible: multiplicative inverses (via the polynomial extended GCD), division, integer powers, and the Frobenius endomorphism a ↦ aᵖ. Irreducibility is a hypothesis Hex.FpPoly.Irreducible carried in the type of every field element, discharged in practice by a checkable Rabin certificate from HexBerlekamp.