When L/K is separable, the trace provides a duality theory via the trace form: the map from L × L to K sending (x, y) to TrL/K(xy) is a nondegenerate, symmetric, bilinear form called the trace form. An example of where this is used is in algebraic number theory in the theory of the different ideal.
The trace form for a finite degree field extension L/K has non-negative signature for any field ordering of K. The converse, that every Witt equivalence class with non-negative signature contains a trace form, is true for algebraic number fields K.
Read more about this topic: Field Trace
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—Ludwig Wittgenstein (1889–1951)