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, NL/K(α), is defined as the determinant of this linear transformation.
If L/K is a Galois extension, one may compute the norm of α ∈ L as the product of all the Galois conjugates of α:
where Gal(L/K) denotes the Galois group of L/K.
For a general field extension L/K, and nonzero α in L, let σ1(α), ..., σn(α) be the roots of the minimal polynomial of α over K (roots listed with multiplicity and lying in some extension field of L); then
If L/K is separable, then each root appears only once in the product (though the exponent, the degree [L:K(α)], may still be greater than 1).
The field norm from the complex numbers to the real numbers sends
- x + iy
- x2 + y2,
because the Galois group of over has two elements, the identity element and complex conjugation, and taking the product yields (x + iy)(x − iy) = x2 + y2.
In this example the norm was the square of the usual Euclidean distance norm in . In general, the field norm is very different from the usual distance norm. We will illustrate that with an example where the field norm can be negative. Consider the number field . The Galois group of over has order d = 2 and is generated by the element which sends to . So the norm of is:
The field norm can also be obtained without the Galois group. Fix a -basis of , say : then multiplication by the number sends 1 to and to . So the determinant of "multiplying by is the determinant of the matrix which sends the vector (corresponding to the first basis element, i.e. 1) to and the vector (which represents the second basis element ) to , viz.:
The determinant of this matrix is −1.
Properties of the norm
Several properties of the norm function hold for any finite extension.
The norm NL/K : L* → K* is a group homomorphism from the multiplicative group of L to the multiplicative group of K, that is
Furthermore, if a in K:
If a ∈ K then
Additionally, the norm behaves well in towers of fields: if M is a finite extension of L, then the norm from M to K is just the composition of the norm from M to L with the norm from L to K, i.e.
Let L = GF(qn) be a finite extension of a finite field K = GF(q). Since L/K is a Galois extension, if α is in L, then the norm of α is the product of all the Galois conjugates of α, i.e.
In this setting we have the additional properties,
The norm of an algebraic integer is again an integer, because it is equal (up to sign) to the constant term of the characteristic 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 OK, the ring of integers of the number field K, N(I) is the number of residue classes in – 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 then N(I) is equal to the absolute value of the norm to Q of α, for α an algebraic integer.
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- Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, CRC Press, ISBN 978-1-4398-7378-6
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