There exists the genus-degree formula for plane, projective, nonsingular curves that relates the (arithmetic) genus of a curve $C\_F$ with the degree of the polynomial $F$ by the following relation:

> 
> $$\frac{1}{2} (\deg F - 1)(\deg F - 2) = g(C\_F)$$
> 
> 
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I know there are a lot of great resources and different ways to proof this. I found the statement in several algebraic geometry or curve theory books (Hartshorne, Milne, Fulton, etc.). My problem is that almost every resource that I have found so far has either only proofed this formula for algebraically closed fields or, if the field was arbitrary, was so scheme-theoretic that I wouldn't even recognise the formula if I saw it.

Two questions:

1. Is this formula even true if the field is not closed (and/or not perfect)?
2. If 1. is true, then what is a (citable) reference that states this formula for curves over arbitrary fields?

I will accept an answer that posts nothing but a citable source (maybe including the page) which contains the statement for arbitrary fields.