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Every field of characteristic 0 contains a copy of the rationals Q and is therefore infinite; every field of characteristic p contains a copy of Zp. There are finite fields and infinite fields of characteristic p.
Every field has a unique smallest subfield, which is called the prime subfield and is contained in every other subfield. For fields of characteristic 0, the prime subfield is isomorphic to Q (the rationals). Fields of characteristic 0 are therefore always infinite. For fields of prime characteristic p, the prime subfield is isomorphic to Zp. Fields of prime characteristic can be either infinite or finite (see Finite field).