Hahn series

In mathematics, Hahn series (sometimes also known as Hahn–Mal'cev–Neumann series) are a type of formal infinite series. They are a generalization of Puiseux series (themselves a generalization of formal power series) and were first introduced by Hans Hahn in 1907 (and then further generalized by Anatoly Maltsev and Bernhard Neumann to a non-commutative setting). They allow for arbitrary exponents of the indeterminate so long as the set supporting them forms a well-ordered subset of the value group (typically or ). Hahn series were first introduced, as groups, in the course of the proof of the Hahn embedding theorem and then studied by him in relation to Hilbert's second problem. The field of Hahn series (in the indeterminate ) over a field and with value group (an ordered group) is the set of formal expressions of the form with such that the support of f is well-ordered. The sum and product of and are given by and (in the latter, the sum over values such that , and is finite because a well-ordered set cannot contain an infinite decreasing sequence). For example, is a Hahn series (over any field) because the set of rationals is well-ordered; it is not a Puiseux series because the denominators in the exponents are unbounded. (And if the base field K has characteristic p, then this Hahn series satisfies the equation so it is algebraic over .) The valuation of a non-zero Hahn series is defined as the smallest such that (in other words, the smallest element of the support of ): this makes into a spherically complete valued field with value group and residue field (justifying a posteriori the terminology). In fact, if has characteristic zero, then is up to (non-unique) isomorphism the only spherically complete valued field with residue field and value group . The valuation defines a topology on . If , then corresponds to an ultrametric absolute value , with respect to which is a complete metric space.