Editing Field trace (section)

==Properties of the trace==
Several properties of the trace function hold for any finite extension.<ref>{{harvnb|Roman|2006|p=151}}</ref>

The trace {{nowrap|Tr{{sub|''L''/''K''}} : ''L'' → ''K''}} is a ''K''-[[linear map]] (a ''K''-linear functional), that is
:<math>\operatorname{Tr}_{L/K}(\alpha a + \beta b) = \alpha \operatorname{Tr}_{L/K}(a)+ \beta \operatorname{Tr}_{L/K}(b) \text{ for all }\alpha, \beta \in K</math>.

If {{nowrap|''α'' ∈ ''K''}} then <math>\operatorname{Tr}_{L/K}(\alpha) = [L:K] \alpha.</math>

Additionally, trace behaves well in [[tower of fields|towers of fields]]: if ''M'' is a finite extension of ''L'', then the trace from ''M'' to ''K'' is just the [[function composition|composition]] of the trace from ''M'' to ''L'' with the trace from ''L'' to ''K'', i.e.
:<math>\operatorname{Tr}_{M/K}=\operatorname{Tr}_{L/K}\circ\operatorname{Tr}_{M/L}</math>.