Field norm

In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield.

Let K be a field and L a finite extension (and hence an algebraic extension) of K. The field L is then a finite dimensional vector space over K. Multiplication by α, an element of L,

is a K-linear transformation of this vector space into itself. The norm, N_L/K(α), is defined as the determinant of this linear transformation.

For nonzero α in L, let σ₁(α), ..., σ_n(α) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of L), then

If L/K is separable then each root appears only once in the product (the exponent [L:K(α)] may still be greater than 1).

More particularly, if L/K is a Galois extension and α is in L, then the norm of α is the product of all the Galois conjugates of α, i.e.

where Gal(L/K) denotes the Galois group of L/K.

The field norm from the complex numbers to the real numbers sends

to

because the Galois group of ${\displaystyle \mathbb {C} }$ over ${\displaystyle \mathbb {R} }$ has two elements, the identity element and complex conjugation, and taking the product yields (x + iy)(x − iy) = x² + y².

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