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, ..., σn 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
Read more about Field Trace: Properties of The Trace, Trace Form
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His belly close to ground. I see the blade,
Blood-stained, continue cutting weeds and shade.”
—Jean Toomer (1894–1967)
“The land of shadows wilt thou trace
And look nor know each other’s face
The present mixed with reasons gone
And past and present all as one
Say maiden can thy life be led
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Then trace thy footsteps on with me
We’re wed to one eternity”
—John Clare (1793–1864)