In my textbook there is the following nameless theorem:
Let $Q=\sum_{i,j=1}^n a_{ij}X_i X_j$ with $a_{ij}=a_{ji}\in K$ be a quadratic form in $n$ variables over a field $K$ not of characteristic $2$. Then there is a linear transformation of coordinates, such that $Q$ has the form $$ Q = \sum_{i=1}^r a_i X_i^2 \text{ with $a_i \neq 0$ for $i = 1, \ldots, r$ } \, .$$ The invariant $r$ is the rank of the quadratic form $Q$.
Is this Sylvester's law of inertia?
I think it must be...
I'm pretty sure you are talking about Hermite reduction. The construction is not the same as Sylvester; we are not making any distinction between positive and negative diagonal elements, indeed a field cannot be ordered unless it is characteristic zero.
NOTE: yes, in George Leo Watson, Integral Quadratic Forms, pages 17-19, he discusses a specific version of this as being Hermite's method of reduction. Earlier on page 9 he mentions a more general version under the heading "Rational diagonalization" where he does not multiply through in order to always have integer entries.
Best with induction on $n.$ Re-order so that $a_{11} \neq 0.$ Consider $$ a_{11}(x_1 + \frac{a_{12}}{a_{11}} x_2 + \frac{a_{13}}{a_{11}} x_3 + \cdots + \frac{a_{1n}}{a_{11}} x_n )^2 $$
This is $$ a_{11} \left( x_1^2 + 2 \frac{a_{12}}{a_{11}} x_1 x_2 + 2 \frac{a_{13}}{a_{11}} x_1 x_3 + \cdots + 2 \frac{a_{1n}}{a_{11}} x_1 x_n + \mbox{ OTHER} \right) $$ or $$ a_{11} x_1^2 + 2 a_{12} x_1 x_2 + 2 a_{13} x_1 x_3 + \cdots + 2 a_{1n} x_1 x_n + \mbox{ OTHER}_2 $$
Now, subtract this off from the original $Q$ you wrote. What remains is a quadratic form in $n-1$ variables. Repeat the first step.
CAVEAT. I need to think about what happens when the only diagonal entries remaining are zero. We just make a trivial change to force the first diagonal entry to be nonzero. For example, consider the form $2xy$ in just two variables. If we take $x = u-v, \; y = u+v$ the new form is $2 u^2 - 2 v^2.$ That is, the ability to force the first diagonal coefficient to be nonzero is simply the ability for the form to represent a nonzero number, that is the form is not the zero form.