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A field with finitely many elements.
Let be a finite field. As is the initial object in the category of rings, there is a unique ring homomorphism , whose regular epi-mono factorization is
here is prime (irreducible) since any subring of a field has no zero divisors. Here is a prime field, usually denoted .
Thus we have an inclusion of fields ; in particular is an -module or vector space, clearly of finite dimension . It follows that has elements.
In addition, since the multiplicative group is cyclic (as shown for example at root of unity), of order , it follows that is a splitting field for the polynomial . As splitting fields are unique up to isomorphism, it follows that up to isomorphism there is just one field of cardinality ; it is denoted .1
The Galois group is a cyclic group of order , generated by the automorphism sending . (One way to see that preserves addition is to write (binomial theorem)
where the second equation follows from the fact that the integer divides the numerator of , but neither factor of the denominator, if . This is true for any commutative algebra over ; see freshman's dream.)
This is called the Frobenius (auto)morphism or Frobenius map. More generally, if divides , then is the fixed field of the automorphism , and is a cyclic group of order that is generated by this automorphism, which is also called the Frobenius map (for the field extension ), or just “the Frobenius” for short.
If is an algebraic closure of , then is the union (a filtered colimit) of the system of such finite field extensions and inclusions between them. If , then may be defined to be the fixed field of the automorphism .
The Galois group is the inverse limit of the system of finite cyclic groups and projections between them; it is isomorphic to the profinite completion
where is the group of -adic integers.