quality:corruption, structural:expansion
Fields

The four fundamental properties of the absolute value for real numbers can be used to generalise the notion of absolute value to an arbitrary field, as follows.

A real-valued function v on a field F is called an absolute value (also a modulus, magnitude, value, or valuation) if it satisfies the following four axioms:

Where 0 denotes the additive identity of F. It follows from positive-definiteness and multiplicativity that v (1) = 1, where 1 denotes the multiplicative identity of F. The real and complex absolute values defined above are examples of absolute values for an arbitrary field.

If v is an absolute value on F, then the function d on F × F, defined by d (a, b) = v (a − b), is a metric and the following are equivalent:

An absolute value which satisfies any (hence all) of the above conditions is said to be non-Archimedean, otherwise it is said to be Archimedean.