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How is the trace pairing function $(x,y) \mapsto Tr(xy)$ on a number field an analogue of the dot product in euclidean space?

(This is a view shared by Keith Conrad and can be found in his notes Discriminants... and The Different Ideal)

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    It's bilinear... it's nondegenerate...2012-05-26
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    Section 3 of the 2nd handout you link to sets up the similarity between lattices in Euclidean space and lattices in number fields. I'm not sure why you are asking after reading that "how" there is an analogue. Did it not come out in Section 3? Beyond the setting of number fields, for any finite *separable* extension of fields $L/K$, the trace pairing $L \times L \rightarrow K$ is perfect, so *every* element of the $K$-dual space of $L$ has the form $f(x) = {\rm Tr}_{L/K}(xy)$ for a unique $y \in L$. Likewise, element of the dual space of ${\mathbf R}^n$ is $f(v) = v\cdot w$ for a unique $w$.2012-06-02

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