Field norm
Encyclopedia
In mathematics
, the (field) norm is a mapping defined in field theory
, to map elements of a larger field into a smaller one.
(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
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
of α, for g in the Galois group
G of L/K.
s to the real numbers sends
to
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 ideal
s. This is done in such a way that if I is an ideal of OK, the ring of integers
of the number field K, N(I) is the number of residue classes in OK/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 αOK there is the expected relation between N(I) and the absolute value
of the norm to Q of α, for α an algebraic integer.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the (field) norm is a mapping defined in field theory
Field theory (mathematics)
Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....
, to map elements of a larger field into a smaller one.
Formal definitions
1. Let K be a field and L a finite extensionField extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
(and hence an algebraic extension
Algebraic extension
In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e...
) of K. Multiplication by α, an element of L, is a K-linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
That is, L is viewed as a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over K, and mα is a linear transformation of this vector space into itself. The norm NL/K(α) is defined as the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
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
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
from the multiplicative group of L to that of K.
2. If L/K is a Galois extension
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
, the norm NL/K of an element α of L is the product of all the conjugates
Conjugate element (field theory)
In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field K, are the roots of the minimal polynomialof α over K.-Example:The cube roots of the number one are:...
- g(α)
of α, for g in the Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
G of L/K.
Example
The field norm from the complex numberComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s 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 integerAlgebraic integer
In number theory, an algebraic integer is a complex number that is a root of some monic polynomial with coefficients in . The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A...
is again an integer, because it is equal (up to sign) to the constant term of the minimal polynomial.
In algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
one defines also norms for ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
s. This is done in such a way that if I is an ideal of OK, the ring of integers
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
of the number field K, N(I) is the number of residue classes in OK/I - i.e. the cardinality of this finite ring
Finite ring
In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements....
. Hence this norm of an ideal is always a positive integer. When I is a principal ideal αOK there is the expected relation between N(I) and the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
of the norm to Q of α, for α an algebraic integer.