Finite Fields and Frobenius

A finite field has prime characteristic p, so it contains a copy of F_p. Viewed as a vector space over this prime field, it has some finite dimension n. Choosing coordinates then counts exactly p to the n elements. Existence goes in the other direction: an irreducible polynomial of degree n over F_p gives a field F_p[x] modulo that polynomial, with unique representatives of degree less than n.

The quotient must use an irreducible polynomial. Otherwise nonzero factors can multiply to zero, preventing inverses. Likewise Z modulo nine has nine elements but is not a field, because the nonzero class of three squares to zero. A field with nine elements instead has characteristic three and dimension two over F_3. Cardinality alone therefore does not tell you which addition and multiplication table to use; the defining relations must accompany every calculation.

In characteristic p, the intermediate binomial coefficients in (a+b) to the p vanish, so Frobenius, sending a to a to the p, preserves addition as well as multiplication. It fixes F_p and is injective because a field has no nonzero nilpotents. On a finite set an injective map is surjective, making Frobenius an automorphism. This argument explains why finiteness is useful: Frobenius need not be onto an arbitrary field of characteristic p.

Every element of F_q, with q = p to the n, satisfies a to the q = a. For nonzero elements this follows from the multiplicative group having q minus one elements; zero satisfies it separately. Thus F_q splits x to the q minus x, whose derivative is minus one, so the polynomial has no repeated roots. Frobenius has order exactly n: a smaller positive order m would make all p to the n elements roots of a polynomial x to the p to the m minus x of smaller degree. It generates the full degree-n Galois group over F_p.

The conjugates of an element a over F_p are its distinct Frobenius iterates. The smallest positive d for which a to the p to the d equals a is the degree of its minimal polynomial. Multiplying the corresponding linear factors produces that polynomial: Frobenius permutes the factors, so it fixes the coefficients, placing them in F_p. This converts an orbit calculation into a polynomial calculation without guessing coefficients.

Inside F_{p^n}, a subfield with p to the d elements exists exactly when d divides n, and it is unique. It consists of the roots of x to the p to the d minus x. Under the correspondence it is fixed by the subgroup generated by the dth iterate of Frobenius. Therefore a field of size p to the twelve has subfields of degrees one, two, three, four, six, and twelve, with inclusions controlled by divisibility rather than numerical size alone.