Galois group of a finite field

Finite fields are the most orderly fields there are, and their Galois theory reflects that: every extension of one finite field by another is cyclic, and there is a single canonical symmetry — raising to the p-th power — whose powers produce all the others. Knowing one map, the Frobenius, tells you the entire symmetry structure.

Precisely, let F_q be the finite field with q = p^m elements (p prime), and consider the extension F_{q^n}/F_q. It is a Galois extension of degree n, and its Galois group is cyclic of order n, generated by the relative Frobenius automorphism phi(x) = x^q. The powers phi, phi^2, ..., phi^n = id are exactly the n distinct automorphisms, and the fixed field of phi^d is F_{q^d} for each divisor d of n.

This gives a complete, transparent Galois correspondence: subfields of F_{q^n} containing F_q correspond to divisors of n, matching the subgroup lattice of the cyclic group Z/nZ. For the absolute version, the Galois group of the algebraic closure of F_p over F_p is the profinite completion of the integers, the inverse limit of the cyclic groups Z/nZ, topologically generated by the Frobenius.