Morphism between algebraic (smooth) curves of degree 1 is an isomorphism. Does the converse holds ? In other words, isomorphism between algebraic smooth curves has always degree 1 ?
Degree of morphism is defined by corresponding extension degree of function field.
I think this does not hold in general, but I don't come up with good example. If this holds in general, I would appreciated if you could tell me in the comment.
Isomorphism of algebraic curves gives isomorphism on the field of rational functions, which is a degree $1$ extension. Smoothness is not required for this implication.