symmetric monoidal (∞,1)-category of spectra
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
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.
There is no ‘canonical’ choice of such a splitting field, just as there is no canonical choice of algebraic closure. So ‘morally’ there is something wrong with saying ‘the’ finite field , although this word usage can be found in the literature. In the special case there is no problem: Given fields containing elements are isomorphic in a unique way. ↩