Field norm

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

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

$m_\alpha:L\to L\;.$

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.

Example

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

x + iy

to

x² + y². This corresponds to the Galois group having two elements, the identity element which sends x + iy to itself, and complex conjugation, which sends x + iy to x - iy.

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.

See also

• Field trace
• Norm of an ideal

References

• Roman, Steven (2006), Field theory, Graduate Texts in Mathematics, 158 (Second ed.), Springer, Chapter 8, ISBN 978-0-387-27677-9