Every field has a distinguished ring homomorphism from the integers, whose kernel determines an important invariant.
Proposition 4.3.1 (Universal map from ). For any field , there is a unique ring homomorphism defined by . Explicitly, (add to itself times for ; negate for ).
Proof. Uniqueness: Any ring homomorphism must satisfy , and since as an additive group, is determined by its value on .
Existence: Define . Then and (verify by induction). Also .∎
The kernel of is a subgroup of , hence of the form for some .
Proposition 4.3.2 (Kernel is zero or prime). The kernel equals where either or is prime.
Proof. Suppose and is not prime, so with . Then In a field, implies or . So or , meaning or . But and , so or is in only if or —contradicting .
Thus if , then must be prime. ∎
Definition 4.3.1 (Characteristic). The characteristic of a field , denoted , is:
if , i.e., is injective;
if for a prime .
Equivalently, means , and is the smallest such positive integer.
Example 4.3.1.
. The map is injective.
for prime . Indeed, in .
.
Note. The field is often denoted and called the prime field of characteristic . Every field of characteristic contains a copy of ; every field of characteristic contains a copy of . Thus and are the “minimal” fields in their respective characteristics.
Theorem 4.3.1 (Existence and uniqueness of finite fields). For every prime and , there exists a unique (up to isomorphism) field with exactly elements, denoted or . These are all the finite fields.
Note. We will not prove this theorem here. The finite field can be constructed as a splitting field of over . There are also infinite fields of characteristic —for example, , the field of rational functions over .