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:
“And in these dark cells,
packed street after street,
souls live, hideous yet
O disfigured, defaced,
with no trace of the beauty
men once held so light.”
—Hilda Doolittle (1886–1961)
“Every neurosis is a primitive form of legal proceeding in which the accused carries on the prosecution, imposes judgment and executes the sentence: all to the end that someone else should not perform the same process.”
—Lionel Trilling (1905–1975)