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
Famous quotes containing the words trace and/or form:
“To love someone is to isolate him from the world, wipe out every trace of him, dispossess him of his shadow, drag him into a murderous future. It is to circle around the other like a dead star and absorb him into a black light.”
—Jean Baudrillard (b. 1929)
“What is a novel if not a conviction of our fellow-men’s existence strong enough to take upon itself a form of imagined life clearer than reality and whose accumulated verisimilitude of selected episodes puts to shame the pride of documentary history?”
—Joseph Conrad (1857–1924)