Let $X$ be a curve, therefore proper , one-dimensional $k$-scheme. Two questions:
How exactly is $K(X)$ defined. I heard it's the fraction field of the stalk at generic point of $X$, but why is this generic point unique? Or is $K(X)$ independent of the choice of the generic point?
Why $K(X)$ has as field extension exactly transcendence degree $1$ over $k$?