Genus and gaps in function fields (Q757480)

Let R and \(R'\) be both function fields over the same constant field K and \(R'\) be an algebraic extension of R of the finite degree \(n=[R' : R]\). The author gives a simple formula for the genus of \(R'\) in terms of the genus of R, n and other data associated with R, e.g. in the cases \(R'=R(y)\), y: \(y^ n=x\), or \(y^ n-y=x\), \(x\in R\). A key role is played by the notion of the gaps at a place of the function field. The methods used in this paper are purely algebraic and the constant field is allowed to be arbitrary (e.g. a finite field).