normsfrom

Normsfrom is a term used in number theory to denote the set of norm values that come from a field extension. Given a finite field extension K/F, the norm map N_{K/F}: K^× → F^× sends each nonzero element of K to the determinant of the F-linear map “multiplication by a” on K. The image N_{K/F}(K^×) is the group of norms from K to F, sometimes simply called the norms from K to F.

Formal definition: For a finite extension K/F, N_{K/F}(α) equals the product of all F-embeddings of K into an algebraic closure applied to α, equivalently the determinant of the F-linear map x ↦ αx. The subset of F^× consisting of all N_{K/F}(α) with α ∈ K^× is a subgroup of F^×, denoted Norms(K/F) or N_{K/F}(K^×).

Examples: If K = C and F = R, N_{C/R}(z) = zȳ = |z|^2, so the norms are the positive real numbers. For finite fields, the norm map N_{F_{q^n}/F_q} is surjective onto F_q^×, so every nonzero element of F_q is a norm from F_{q^n}.

Applications: Norms from extensions are central in class field theory, Brauer groups, and local-global principles such as the Hasse norm theorem, which relates global norms to local norms. They also underpin explicit constructions in arithmetic geometry.

Computation: In practice, the norm can be computed as the determinant of the F-linear map “multiplication by a” or via field embeddings; many computer algebra systems provide functions to compute N_{K/F}(α) or the entire Norms(K/F).