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 mα is a linear transformation of this vector space into itself. The norm NL/K(α) is defined as the determinant of this linear transformation. Properties of the determinant imply that the norm belongs to K and
- NL/K(αβ) = NL/K(α)NL/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 NL/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.
Read more about this topic: Field Norm
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