prime field

Every field, no matter how large or exotic, has a smallest possible field hiding inside it — the one you cannot avoid, generated just by the element 1 and the field operations. This irreducible core is the prime field. It is the same for all fields sharing a given characteristic, so it is the unchangeable bedrock on which the rest of the field is built. Nothing smaller is a field at all.

Precisely, the prime field of K is the intersection of all subfields of K, equivalently the smallest subfield, equivalently the subfield generated by the multiplicative identity 1. Its isomorphism type is dictated entirely by the characteristic: if char(K) = 0 the prime field is isomorphic to the rationals Q, and if char(K) = p the prime field is isomorphic to the finite field F_p of p elements.

Because the prime field is canonically embedded, every field is automatically an extension of either Q or some F_p, and any field homomorphism must fix the prime field pointwise (since it must send 1 to 1). This makes the prime field the natural ground over which to do field theory, and it is why characteristic 0 fields and characteristic p fields are studied as two fundamentally separate families.