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Concept
Field norm
Summary
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.
Official source
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