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The mapping of a field into a field (where is a finite extension of ) that sends an element to the trace of the matrix (cf. Trace of a square matrix) of the -linear mapping sending to . is a homomorphism of the additive groups.

If is a separable extension, then

where the sum is taken over all -isomorphisms of into an algebraic closure of . The trace mapping is transitive, that is, if and are finite extensions, then for any ,


Comments

Especially in older mathematical literature, instead of one finds (from the German "Spur" ) as notation for this mapping.

References

[a1] N. Jacobson, "Lectures in abstract algebra" , 3. Theory of fields and Galois theory , Springer, reprint (1975)
[a2] N. Jacobson, "Basic algebra" , 1 , Freeman (1985)
[a3] S. Lang, "Algebra" , Addison-Wesley (1965)
How to Cite This Entry:
Trace. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trace&oldid=35824
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article