Every ring receives a distinguished homomorphism from the integers. For a field, its kernel is an important invariant.
For an element of a ring and , we write for the -th multiple of in the additive group: ( terms) for , , and for . This is the additive version of the powers in a group from Chapter 2, so and . Distributivity and Proposition 4.1.3 give .
Proposition 4.4.1 (The homomorphism from ). For any ring there is a unique ring homomorphism , namely .
Proof. Uniqueness: a ring homomorphism is in particular a homomorphism of additive groups with . Since generates , it is determined by this value: .
Existence: define . Then and . Finally, ∎
Now let be a field and this homomorphism. Its kernel is a subgroup of , hence equal to for a unique (Theorem 2.4.5).
Proposition 4.4.2 (The kernel is zero or prime). The kernel of is where either or is prime.
Proof. We have , since . Suppose is not prime, so with . Then and since a field has no zero divisors, or . That is, or , which is impossible for . Hence if , then is prime. ∎
Definition 4.4.3 (Characteristic). The characteristic of a field , denoted , is
- if , i.e., is injective;
- if for a prime .
Equivalently, means that and is the smallest positive integer with this property, while means that for every .
In a field of characteristic , we have for every . In characteristic in particular, , that is, .
Example 4.4.4.
- : the map is the inclusion.
- for every prime : in , while for .
- : these fields contain as a subfield, so the homomorphism from to either of them is followed by the inclusion, and has the same kernel.
Note. The field is the prime field of characteristic . Every field of characteristic contains a copy of : since exactly when divides , the map , , is an injective ring homomorphism. Every field of characteristic contains a copy of , via . Thus and are the smallest fields of their characteristics.
Theorem 4.4.5 (Existence and uniqueness of finite fields). For every prime and every , there exists a field with exactly elements, unique up to isomorphism, denoted or . Every finite field is of this form.
Note. We do not prove this theorem here. The field can be constructed as a splitting field of over (splitting fields are discussed in Chapter 17). There are also infinite fields of characteristic , for example the field of rational functions and the field of Laurent series .