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Let $E/F$ be a finite extension, then we get a quadratic form $\alpha\mapsto Tr_F^E(\alpha^2),\alpha\in E$ over $F$. Is there any invariant in number theory that's equivalent to this quadratic form?

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This quadratic form, the "trace form", and the corresponding bilinear form $(x, y) \mapsto \operatorname{Tr}^E_F(xy)$, is an important tool in algebraic number theory (e.g. it's vital in understanding ramification, since it's used in the definition of the different ideal). I'm not sure why you want to find some other invariant that's equivalent to this one: what's wrong with the one that you've got?