Algebraic Geometry
In the context of algebraic geometry, the notion of branch points can be generalized to mappings between arbitrary algebraic curves. Let ƒ:X → Y be a morphism of algebraic curves. By pulling back rational functions on Y to rational functions on X, K(X) is a field extension of K(Y). The degree of ƒ is defined to be the degree of this field extension, and ƒ is said to be finite if the degree is finite.
Assume that ƒ is finite. For a point P ∈ X, the ramification index eP is defined as follows. Let Q = ƒ(P) and let t be a local uniformizing parameter at P; that is, t is a regular function defined in a neighborhood of Q with t(Q) = 0 whose differential is nonzero. Pulling back t by ƒ defines a regular function on X. Then
where vP is the valuation in the local ring of regular functions at P. That is, eP is the order to which vanishes at P. If eP > 1, then ƒ is said to be ramified at P. In that case, Q is called a branch point.
Read more about this topic: Branch Point
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