The Kolchin Seminar in Differential Algebra Finitely Generated Difference Field Extensions

Let K be a difference field of zero characteristic with a basic set σ = {α1,..., αn}, that is, a field K considered together with the mutually commuting injective endomorphisms {α1,..., αn} of K. In this case K is also called a σ-field. If αi are automorphisms, the σ-field K is called inversive. It is called ordinary if n = 1, and partial if n> 1. If K0 is a subfield of K and α(a) ∈ K0 for any a ∈ K0, α ∈ σ, we say that K0 is a difference (or σ-) subfield of K and K is a difference (σ-) field extension of K0. We also say that we have a σ-field extension K/K0. If B ⊆ K, then the intersection of all σ-subfields of K containing K0 and B is called the difference (σ-) subfield of K generated by the set B over K0; it is denoted by K0〈B〉. As a field, K0〈B 〉 = K0({τ(b)|b ∈ B, τ ∈ T}). The set B is called the set of σ-generators of K/K0. If |B | <∞, B = {b1,..., bk}, we say that K is a finitely generated difference (σ-) field extension of K0 and write K = K0〈b1,..., bk〉. The concept of limit degree of a finitely generated ordinary difference field extension was introduced by R. Cohn (1956) as follows. Let K be an ordinary difference ring with a basic set σ = {α} and L = K〈S 〉 a σ-fiel