Valuations, primes and irreducibility in polynomial rings and rational function fields

Abstract:

The set of extensions of the valuation v on a linearly compact (i.e. maximal) field F to the polynomial ring $F[x]$ is shown to depend only on the value group and residue class field of v. By a method related to Mac Lane’s construction of (rank one) valuations on polynomial rings, a determining invariant is associated with each such extension, called its “signature". Very roughly, a signature is a pair of sequences, one in the algebraic closure of the residue class field of v and one in the divisible closure of the value group of v. Signatures are also associated with various mathematical objects by means of the extensions of the above sort which naturally arise from them. For example, the set of nonconstant monic irreducible polynomials in $F[x]$, the set of all finite Harrison primes of the polynomial ring of a global field, and the set of equivalence classes of valuations on the field of rational functions over a global field are each shown to be bijective with a simple set of signatures. Moreover, these objects are studied by means of their associated signatures. For example, necessary and sufficient conditions for irreducibility in $F[x]$ are given, independent of the language of signatures.