Norm form
Encyclopedia
In mathematics
, a norm form is a homogeneous form in n variables constructed from the field norm
of a field extension
L/K of degree n. That is, writing N for the norm mapping to K, and selecting a basis
for L as a vector space over K, the form is given by
in variables
In number theory
norm forms are studied as Diophantine equation
s, where they generalize, for example, the Pell equation. For this application the field K is usually the rational number field, the field L is an algebraic number field
, and the basis is taken of some order in the ring of integers
OL of L.
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...
, a norm form is a homogeneous form in n variables constructed from the field norm
Field norm
In mathematics, the norm is a mapping defined in field theory, to map elements of a larger field into a smaller one.-Formal definitions:1. Let K be a field and L a finite extension of K...
of a field extension
Field 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...
L/K of degree n. That is, writing N for the norm mapping to K, and selecting a basis
- e1, ..., en
for L as a vector space over K, the form is given by
- N(x1e1 + ... + xnen)
in variables
- x1, ..., xn.
In number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
norm forms are studied as Diophantine equation
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
s, where they generalize, for example, the Pell equation. For this application the field K is usually the rational number field, the field L is an algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
, and the basis is taken of some order in 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...
OL of L.