Characteristic (algebra) - Case of Fields

Case of Fields

As mentioned above, the characteristic of any field is either 0 or a prime number. A field of non-zero characteristic is called a field of finite characteristic or a field of positive characteristic.

For any field F, there is a minimal subfield, namely the prime field, the smallest subfield containing 1_F. It is isomorphic either to the rational number field Q, or a finite field of prime order, F_p; the structure of the prime field and the characteristic each determine the other. Fields of characteristic zero have the most familiar properties; for practical purposes they resemble subfields of the complex numbers (unless they have very large cardinality, that is; in fact, any field of characteristic zero and cardinality at most continuum is isomorphic to a subfield of complex numbers). The p-adic fields or any finite extension of them are characteristic zero fields, much applied in number theory, that are constructed from rings of characteristic pk, as k → ∞.

For any ordered field, as the field of rational numbers Q or the field of real numbers R, the characteristic is 0. Thus, number fields and the field of complex numbers C are of characteristic zero. Actually, every field of characteristic zero is the quotient field of a ring Q/P where X is a set of variables and P a set of polynomials in Q. The finite field GF(pn) has characteristic p. There exist infinite fields of prime characteristic. For example, the field of all rational functions over Z/pZ, the algebraic closure of Z/pZ or the field of formal Laurent series Z/pZ((T)).

The size of any finite ring of prime characteristic p is a power of p. Since in that case it must contain Z/pZ it must also be a vector space over that field and from linear algebra we know that the sizes of finite vector spaces over finite fields are a power of the size of the field. This also shows that the size of any finite vector space is a prime power. (It is a vector space over a finite field, which we have shown to be of size pn. So its size is (pn)m = pnm.)