Field Norm - Formal Definitions

Formal Definitions

1. Let K be a field and L a finite extension (and hence an algebraic extension) of K. Multiplication by α, an element of L, is a K-linear transformation

That is, L is viewed as a vector space over K, and m_α is a linear transformation of this vector space into itself. The norm N_L/K(α) is defined as the determinant of this linear transformation. Properties of the determinant imply that the norm belongs to K and

N_L/K(αβ) = N_L/K(α)N_L/K(β)

so that the norm, when considered on non-zero elements, is a group homomorphism from the multiplicative group of L to that of K.

2. If L/K is a Galois extension, the norm N_L/K of an element α of L is the product of all the conjugates

g(α)

of α, for g in the Galois group G of L/K.