Let be a field. A valuation or absolute value on is a function satisfying the properties:
for all , with equality if and only if
If a valuation satisfies , then we say that it is a non-archimedean valuation. Otherwise we say that it is an archimedean valuation.
Every valuation on defines a metric on , given by . This metric is an ultrametric if and only if the valuation is non-archimedean. Two valuations are equivalent if their corresponding metrics induce the same topology on . An equivalence class of valuations on is called a prime of . If consists of archimedean valuations, we say that is an infinite prime, or archimedean prime. Otherwise, we say that is a finite prime, or non-archimedean prime.
In the case where is a number field, primes as defined above generalize the notion of prime ideals in the following way. Let be a nonzero prime ideal11By “prime ideal” we mean “prime fractional ideal of ” or equivalently “prime ideal of the ring of integers of ”. We do not mean literally a prime ideal of the ring , which would be the zero ideal., considered as a fractional ideal. For every nonzero element , let be the unique integer such that but . Define
where denotes the absolute norm of . Then is a non–archimedean valuation on , and furthermore every non-archimedean valuation on is equivalent to for some prime ideal . Hence, the prime ideals of correspond bijectively with the finite primes of , and it is in this sense that the notion of primes as valuations generalizes that of a prime ideal.
As for the archimedean valuations, when is a number field every embedding of into or yields a valuation of by way of the standard absolute value on or , and one can show that every archimedean valuation of is equivalent to one arising in this way. Thus the infinite primes of correspond to embeddings of into or . Such a prime is called real or complex according to whether the valuations comprising it arise from real or complex embeddings.
|Date of creation||2013-03-22 12:35:07|
|Last modified on||2013-03-22 12:35:07|
|Last modified by||djao (24)|