Traces and norms
In §§ 1–3, we will consider exclusively local fields (assumed to be commutative). We denote by K a local field and by K′ an algebraic extension of K of finite degree n over K. If K is an R-field and K′ ≠ K, we must have K = R, K′ = C, n = 2; then, by corollary 3 of prop. 4, Chap. III-3, Tr C / R (x) = x+x̄ and N C / R (x) = xx̄; Tr C / R maps C onto R, and N C / R maps C x onto R + x , which is a subgroup of R x of index 2.
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