After reading Fulton's book "Algebraic Curves", I know how to do resolution of singular points on curves. Given an affine equation, I can get it's non-singular affine model, i.e the normalization of its affine coordinate ring. The problem is that in Fulton's book, he worked with algebraically closed field and he worked with the origin (0,0) also. If one do blowing-up (p.165 in Fulton's book) of a curve defined a field $k$ which is not necessary algebraically colsed and with a singular point which is not necessary $k$-rational, one get an eqaution which is defined over a finite extension of the field $k$. But for any affine curve (i.e integral scheme of dimension 1), one can form the normalization of its coordinate ring and hence get a non-singular affine curve which is birational to the original curve. My question is that how one can get the equation for this normalized affine curve? I know there are algorithms to compute the integral closure of function fields, but the result is an integral basis rather that an equation. Moreover, I would like to know if there is a systematic method to get this normalized affine equation with the blowing-up method.
It won't be possible in general to get a single equation, because the curve does not necessarily admit a nonsingular plane model (globally). It is possible to get equations using Computer Algebra Systems, for instance Singular does it as explained here.
Concerning blowups: if the curve has a singular point which is not rational, then all of its conjugate points will be singular too, and they'll form a scheme that is defined over $k$. I never had to deal with such examples, but if $k$ is perfect I suppose that the blowup centered at that scheme will simplify the singularity and, iterating, eventually resolve it.