Absolute Value (algebra) - Completions

Completions

Given an integral domain D with an absolute value, we can define the Cauchy sequences of elements of D with respect to the absolute value by requiring that for every r > 0 there is a positive integer N such that for all integers m, n > N one has | x_m − x_n | < r. It is not hard to show that Cauchy sequences under pointwise addition and multiplication form a ring. One can also define null sequences as sequences of elements of D such that | a_n | converges to zero. Null sequences are a prime ideal in the ring of Cauchy sequences, and the quotient ring is therefore an integral domain. The domain D is embedded in this quotient ring, called the completion of D with respect to the absolute value | x |.

Since fields are integral domains, this is also a construction for the completion of a field with respect to an absolute value. To show that the result is a field, and not just an integral domain, we can either show that null sequences form a maximal ideal, or else construct the inverse directly. The latter can be easily done by taking, for all nonzero elements of the quotient ring, a sequence starting from a point beyond the last zero element of the sequence. Any nonzero element of the quotient ring will differ by a null sequence from such a sequence, and by taking pointwise inversion we can find a representative inverse element.

Another theorem of Alexander Ostrowski has it that any field complete with respect to an Archimedean absolute value is isomorphic to either the real or the complex numbers and the valuation is equivalent to the usual one. The Gelfand-Tornheim theorem states that any field with an Archimedean valuation is isomorphic to a subfield of C, the valuation being equivalent to the usual absolute value on C.