Let K be a field and S a ring. Morphisms can be used to construct subfields.
Theorem
Suppose that f : K -> S is a ring morphism.
- The morphism f is injective.
- The image of f is a subring of S isomorphic to K.
- If S = K, then
{x
K |
f(x) = x} is a subfield of
K.
The subfield in Part 3 is called the fixed
field of f. A fixed point of f is an element
x K such that f(x)
= x. Thus, the fixed field of f consists of all fixed
points of f.
We apply the above result to the case where K has characteristic p > 0. Let L be the smallest subfield of K (isomorphic to Z/pZ).
Proposition
Let q be a power of p.
-
(x + y)q = xq
+ yq
for all x, y
K;
- the map x -> xq is a morphism K -> K;
- for each g L[X], we have
g(X)p =
g(Xp);
- the subset
{x K | xq =
x}
is a
finite subfield of K;
- {x
K | xp = x}
= L.