Field norm

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In mathematics, the (field) norm is a mapping defined in field theory, to map elements of a larger field into a smaller one.

1 Formal definitions
2 Example
3 Further properties
4 See also

[edit] Formal definitions

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

$m_\alpha:L\to L$

The norm N_L/K(α) is defined as the determinant of m_α. 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.

3. The norm of an algebraic element α over K can be defined as the product N(α) of the roots of its minimal polynomial, which are different pairwise since the extension is Galois and so the minimal polynomial is separable. Assuming α is in L, the elements

g(α)

are those roots, each repeated a certain number d of times. Here

d = [L: M]

is the degree of L over the subfield M of L that is the splitting field of the minimal polynomial of α. Therefore the relationship of the norms is

N_L/K(α) = N(α)^d.

[edit] Example

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

x + iy

x² + y².

[edit] Further properties

The norm of an algebraic integer is again an integer, because it is equal (up to sign) to the constant term of the minimal polynomial.

In algebraic number theory one defines also norms for ideals. This is done in such a way that if I is an ideal of O_K, the ring of integers of the number field K, N(I) is the number of residue classes in O_K/I - i.e. the cardinality of this finite ring. Hence this norm of an ideal is always a positive integer. When I is a principal ideal αO_K there is the expected relation between N(I) and the absolute value of the norm to Q of α, for α an algebraic integer.