How easy is to get hepatitis C name “difference field” originated from this example: an equation of the form , where f is the unknown function to be found and P is a polynomial over K, is called an algebraic difference equation. One can replace K by other fields, e.g., the field of meromorphic functions on C or on R.Let K be a field, hepatitis c transmission its separable closure and . The is a difference field. Note that because the algebraic closure Kalg of K is purely inseparable over Ks, σ extends uniquely to an automorphism of Kalg . One often identifies Gal.
The structures described above are a particular example. More generally, we have:
Let K be a perfect field of characteristic, and q a power of p. Then is a difference field. If the field K is algebraically closed then
This is because for fixed q the map is definable in the language of fields, and because the theory of algebraically closed fields ismodel complete.
Definitions, notation and some basic algebraic results. In the literature, a difference field is a field K with a distinguished hepatitis hepatitis c transmission Cis onto, then is called an inversive difference field. However, a simple inductive limit argument shows that every difference field has a unique (up to isomorphism) inversive closure. We will assume in what follows hepatitis C, that all our difference fields are inversive hepatitis c transmission. The references are to [Cohn 1965].
Let K be a difference field, and let be indeterminates a difference polynomial over is an ordinary polynomial with coefficients in K, in the variables. The ring of those difference polynomials is denotedand σ extends naturally to , in the way suggested by the names of the variables.As defined, is not onto. It is sometimes convenient to consider.