Fields and Integral Domains
If D is an integral domain with absolute value | x |, then we may extend the definition of the absolute value to the field of fractions of D by setting
On the other hand, if F is a field with ultrametric absolute value | x |, then the set of elements of F such that | x | ≤ 1 defines a valuation ring, which is a subring D of F such that for every nonzero element x of F, at least one of x or x−1 belongs to D. Since F is a field, D has no zero divisors and is an integral domain. It has a unique maximal ideal consisting of all x such that | x | < 1, and is therefore a local ring.
Read more about this topic: Absolute Value (algebra)
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