Field Trace

In mathematics, the field trace is a function defined with respect to a finite field extension L/K. It is a K-linear map from L to K. As an example, if L/K is a Galois extension and α is in L, then the trace of α is the sum of all the Galois conjugates of α, i.e.

where Gal(L/K) denotes the Galois group of L/K.

For a general finite extension L/K, the trace of an element α can be defined as the trace of the K-linear map "multiplication by α", that is, the map from L to itself sending x to αx. If L/K is inseparable, then the trace map is identically 0.

When L/K is separable, a formula similar to the Galois case above can be obtained. If σ₁, ..., σ_n are the distinct K-linear field embeddings of L into an algebraically closed field F containing K (where n is the degree of the extension L/K), then