A quadratic form is a map F^n ---> F. The signature is supposed to be independent of the choice of basis. So the number of positve/negative eigenvalues makes sense only for F = the reals. For instance in the complex numbers you can always find an orthogonal basis with gram matrix diag(1,....,1,0,...0). So here the invariant would be the rank. If you want a signature over another field, you have to fix an embedding F--> RR and consider the quadratic form as a real quadratic form.
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