Field norm

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. (Note that there may be a repetition in the terms of the product.) For a general field extension L/K, and nonzero α in L, let σ_1(α), ..., σ_n(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). One of the basic examples of norms comes from quadratic field extensions where is a square-free integer. Then, the multiplication map by on an element is The element can be represented by the vector since there is a direct sum decomposition as a -vector space. The matrix of is then and the norm is , since it is the determinant of this matrix. Consider the number field . The Galois group of over has order 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 , (corresponding to the second basis element, i.e., ) to , viz.: The determinant of this matrix is −1. Another easy class of examples comes from field extensions of the form where the prime factorization of contains no -th powers, for a fixed odd prime.