Let
F be a
field. The intersection of all
subfields
of
F is itself a subfield, called the
prime subfield
of
F.
Proposition The prime subfield of F is either
isomorphic to Q or Zp,
for a prime p.
Definition We say that F has characteristic zero if it has
prime subfield isomorphic to Q and characteristic p if
it has prime subfield isomorphic to Zp.
Proof of the proposition:
Let P be the prime subfield. Since it is a subfield of F
it contains 1F.
We need some notation.
If n is a positive integer define
n.1F=1F+....+1F
for n copies of 1F added together.
We define -n.1F=-(n.1F) and
0.1F=0F.
With this notation it is easy to check that
n.1F+m.1F=(n+m).1F
(n.1F)(m.1F)=(nm).1F
There is a function f:Z-->P defined by
f(n)=n.1F. The two previous formulae show that
f is a ring homomorphism. The kernel is an ideal
of Z. Since Z is a principal ideal domain
there exists a (nonnegative) integer n such that
the kernel consists of all multiples of n.
There are two possibilities. Either
-
n>0
-
n=0
Consider case 1. first.
Note that we cannot have
n=1 since we cannot have
1F=0F.
Now I claim that
n must be prime.
For if
n=rs, with
r,s> 1, then we can think
about
(r.1F)(s.1F)=n.1F=0
By the choice of
n we have that
r.1F and
s.1F are nonzero.
Since
P is a field
r.1F is therefore
a
unit. Multiplying the equation by its inverse we deduce that
s.1F=0. This contradiction shows that
n=p
is prime.
Thus by the first isomorphism theorem the image of f is
isomorphic to Zp. The image of f is therefore
a field. Since it is a subring of F contained in P
it must coincide with P, proving one part of the proposition.
Consider case 2. Since the kernel is {0} f is injective.
Consider the field of fractions of the image of f as a subring
of F. Clearly this coincides with P and is isomorphic to Q.
Examples
-
Q,R,C
all have characteristic zero, as does any subfield
of C.
-
A finite field has characteristic p, for a prime p
(there isn't room in a finite field for Q).
But not all fields of finite characteristic are finite.
For example, consider rational functions in one variable K(x)
over a finite field. This field has finite characteristic, but infinitely
many elements. Another example is the algebraic closure of a finite field.