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Short version:

Over which fields is the (appropriate version of the) "Sylvester law of inertia" valid?

Long version:

Let $V$ be a finite dimensional vector space over the field $\Bbbk$ of characteristic different from $2$ (so quadratic forms are the same as symmetric bilinear forms). Consider the "discriminant" function

$$\mathrm{discr}: \mathrm{Sym}^2(V^*)\to \Bbbk/(\Bbbk^{\times})^2,$$Bbbk/\Bbbk^2,$$ where \Bbbk/(\Bbbk^{\times})^2 \Bbbk/\Bbbk^2 is the quotient of the multiplicative monoid \Bbbk by the invertible squares, which maps a (possibly degenerate) symmetric bilinear form q to the determinant of it's matrix relative to any base, which is well defined up to multiplication by squares. Of course, this is not a complete invariant, as different degenerate quadratic forms all have discriminant 0. Given q\in\mathrm{Sym}^2(V^*), and an orthogonal basis B for V in which q has the form \Sigma_i\lambda_ix_iy_i, we have a "signature" map $$\sigma_q^B:\Bbbk/(\Bbbk^{\times})^2\to\mathbb{N}$$\sigma_q^B:\Bbbk/\Bbbk^2\to\mathbb{N}$$