Valuations on Extensions of Weyl Skew Fields Original Research Article

Letksubset ofKbe a finitely generated field extension of transcendence degree 1. Consider the corresponding extension image1(k)subset ofimage1(K) of skew fields of the first Weyl algebra. We show that the image1(k)-valuations of image1(K) are in one-to-one correspondence with thek-valuations ofK. The intersection of the image1(k)-valuation rings of image1(K) isnotimage1(k) as one would expect but a principal ideal domain such that its quotient division ring is image1(K). IfDis a positive divisor, then image(D)={fset membership, variantimage1(K) : div(f)+D≥0} is a finite-dimensional vector space over image1(k) whose dimension equals the degree ofD.