Any ring of characteristic is infinite. The ring of integers modulo has characteristic . If is a subring of , then and have the same characteristic. For example, if is prime and is an irreducible polynomial with coefficients in the field with elements, then the quotient ring is a field of characteristic . Another example: The field of complex numbers contains , so the characteristic of is . A -algebra is equivalently a ring whose characteristic divides . This is because for every ring there is a ring homomorphism , and this map factors through if and only if the characteristic of divides . In this case for any in the ring, then adding to itself times gives . If a commutative ring has prime characteristic , then we have for all elements and in – the normally incorrect "freshman's dream" holds for power . The map then defines a ring homomorphism It is called the Frobenius homomorphism. If is an integral domain it is injective. Case of fields As mentioned above, the characteristic of any field is either or a prime number. A field of non-zero characteristic is called a field of finite characteristic or positive characteristic or prime characteristic. The characteristic exponent is defined similarly, except that it is equal to if the characteristic is ; otherwise it has the same value as the characteristic. Any field has a unique minimal subfield, also called its . This subfield is isomorphic to either the rational number field or a finite field of prime order. Two prime fields of the same characteristic are isomorphic, and this isomorphism is unique. In other words, there is essentially a unique prime field in each characteristic. Fields of characteristic zero The most common fields of characteristic zero are the subfields of the complex numbers. The p-adic fields are characteristic zero fields that are widely used in number theory. They have absolute values which are very different from those of complex numbers. For any ordered field, such as the field of rational numbers or the field of real numbers , the characteristic is . Thus, every algebraic number field and the field of complex numbers are of characteristic zero. Fields of prime characteristic The finite field GF() has characteristic p. There exist infinite fields of prime characteristic. For example, the field of all rational functions over , the algebraic closure of or the field of