About the integral closure of a DVR inside a galois extension of function field with two variables

Edit: I see my mistake now (murphy), w(X + Y) = min(w(X), w(Y)) only when the valuations are not the same.

I have the following algebraic problem, which I encountred after thinking about blowups.

My problem is as follows:

let $$k$$ be a field, and let $$k(X, Y)$$ be the field or rational functions over $$k$$ in $$X,Y$$. letting $$S_1 = S_1(X,Y) = X + Y$$ and $$S_2 = S_2(X,Y) = X Y$$ be the symmetric polynomials in $$X,Y$$. we can form the subfield $$k(S_1, S_2) \subset k(X,Y)$$. Now, let $$R = k(\frac{S_2}{S_1})[S_1]_\mathfrak{p} \subset k(S_1,S_2)$$ where $$\mathfrak{p} = (S_1)$$ is the ideal generated by $$S_1$$. (i.e. $$R$$ is the localization of $$k(\frac{S_2}{S_1})[S_1]$$ at $$\mathfrak{p}$$) and let $$A$$ be its integral clousre inside $$k(X,Y)$$. (elements of $$A$$ are elements of $$k(X,Y)$$ which are integral over $$R$$) we have the following image: $$\require{AMScd}$$ $$\begin{CD} A @>>> k(X,Y)\\ @AAA @AAA\\ R @>>> k(S_1, S_2) \end{CD}$$

Now, I have a good reason to believe that:

1. R is a DVR, with uniformizator $$S_1$$ (and also $$S_2$$)
2. A is a DVR - as the integral closure of a DVR inside a finite (even galois) field extension.(with valuation $$w$$ that extends the valuation of $$R$$

$$X, Y$$ are roots of the polynomial $$(T-X)(T-Y) = T^2 - S_1T + S_2$$ which is integral over $$R$$, hence $$X,Y$$ are in $$A$$. since $$S_1$$ and $$S_2$$ differ by a unit, they share the same valuation under $$w$$ and we get the following set of equalities:

$$min(w(X), W(Y)) = w(S_1) = w(S_2) = w(X) + w(Y)$$ which implies either that $$w(X) = w(Y) = 0$$ or that one of them is $$-\infty$$ both are impossible.

What am I missing?

I encountred the problem after thinking about the following picture: let $$C = A_k^1$$. we have a morphism $$C^2 \rightarrow C^{(2)}$$ where $$C^{(2)}$$ is the symmetric product.

blowing up the symmetric product at the point $$P=(0,0)$$ we get the following diagram:

$$\require{AMScd}$$ $$\begin{CD} Z @>>> Bl_{(0,0)}(C^{(2)})\\ @VVV @VVV\\ P @>>> C^{(2)} \end{CD}$$

the local ring at the generic point of $$Z$$ is $$R$$, which lives inside the function field of $$C^{(2)}$$. (this is also why I believe it is indeed a DVR, because $$Z$$ is of codim 1) and A is the integral closure inside the function field of $$C^{2}$$

The above is obviously a contradiction, and after thinking for a while, I can't find my mistake, and would love to get some help :)

$$w(S_2)=w(S_1) = 1$$
Both $$X,Y$$ are roots of $$T^2-S_1 T+u S_1$$ with $$u=S_2/S_1$$, this polynomial is Eisenstein at $$(S_1)$$ so $$w(X)=w(Y)=1/2$$