Proving normality of affine schemes

One of the exercises in Ravi Vakil's algebraic geometry notes, Ex. $5.4.$I(b), is to show that $$ \operatorname{Spec}\left(k[x_1, \ldots, x_n]/(x_1^2 + \cdots + x_m^2)\right) $$ is normal, where $k$ is any field of $\operatorname{char}(k)\neq 2$, and $n \geq m \geq 3$.

I have absolutely no idea how to get started on this. Is there anyone that could give a hint as to how one would approach this problem?

Solution 1:

The following steps lead to a solution:

Step 1: Suppose the characteristic of a field $k$ is not $2$. If $f \in k[x_1,\ldots,x_n]$ is square-free and non-constant, then $A = k[x_1,\ldots,x_n][z]/(z^2 - f)$ is integrally closed.

Hint for step 1: Follow the method that I employ in my answer here.

Step 2: Show that the ring in your question above is a domain and then show it is integrally closed using step 1. Hint for showing that it is a domain: Dehomogenize with respect to the last variable and apply Eisenstein's Criterion.

Step 3: Conclude your result.