Field F_p - Knowlpedia

For a prime $p$ , the field $\mathbb F_p$ is the quotient field $\mathbb Z/p\mathbb Z$ , with residue-class addition and multiplication.

Operations and maps

Every nonzero residue has an inverse because p is prime. A unit-preserving field endomorphism fixes $1$ , hence every sum of copies of $1$ , and is therefore the identity. The zero map is allowed as a nonunital ring map, but never as a field map.

Underlying groups

Its additive group is cyclic of order $p$ ; its nonzero elements form a separate multiplicative group of order $p-1$ . These group structures must be selected explicitly before taking group Hom-sets.