Field norm - Citizendia

In mathematics, the (field) norm is a mapping defined in field theory, to map elements of a larger field into a smaller one. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and

1 Formal definitions
2 Example
3 Further properties
4 See also

Formal definitions

1. Let K be a field and L a finite algebraic extension of K. In Abstract algebra, a Field extension L / K is called algebraic if every element of L is algebraic over K, i Multiplication by α, an element of L, is a K-linear transformation

$m_\alpha:L\to L$

The norm N_L/K(α) is defined as the determinant of m_α. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n 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. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function

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. In Mathematics, a Galois extension is an algebraic field extension E / F satisfying certain conditions (described below one also says that the A conjugate root is a root of a Polynomial function that has a Conjugate which is also a root of the polynomial function In Mathematics, a Galois group is a group associated with a certain type of Field extension.

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. In Linear algebra, the minimal polynomial of an n -by- n matrix A over a field F is the Monic polynomial In Mathematics, a Polynomial P ( X) is separable over a field K if all of its irreducible factors have Distinct 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. In Abstract algebra, the splitting field of a Polynomial P ( X) over a given field K is a Field extension Therefore the relationship of the norms is

N_L/K(α) = N(α)^d.

Example

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

x + iy

x² + y². Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the real numbers may be described informally in several different ways

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. This article deals with the ring of complex numbers integral over Z.

In algebraic number theory one defines also norms for ideals. In Mathematics, algebraic number theory is a major branch of Number theory which studies the Algebraic structures related to Algebraic integers In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. 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. In Mathematics, the ring of integers is the set of Integers made an Algebraic structure Z with the operations of integer addition In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the 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. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

Contents

Formal definitions

Example

Further properties

See also