A RELATION BETWEEN GALOIS AUTOMORPHISM AND CURVE SINGULARITY
Let ?be an algebraic function field of one variable over an algebraically closed field kof characteristic zero. Suppose ?is a Galois extension of ?and let G be the Galois group. Then, naturally the following basic question arises: How can we express ?as an element of ?for ?We consider this question using a geometric point of view. Let ?be an irreducible polynomial defining ?and C be the projective plane curve defined by ?Suppose ?Then, we have a partial answer: if Chas a singular point of multiplicity ?then ?can be expressed as a linear fractional function ? ?with respect to y.
Galois extension, automorphism, function field.