In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a norm form is a
homogeneous form
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
in ''n'' variables constructed from the
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.
Formal definition
Let ''K'' be a field and ''L'' a finite extension (and hence an algebraic extension) of ''K ...
of a
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
''L''/''K'' of degree ''n''. That is, writing ''N'' for the norm mapping to ''K'', and selecting a basis ''e''
1, ..., ''e''
''n'' for ''L'' as a vector space over ''K'', the form is given by
:''N''(''x''
1''e''
1 + ... + ''x''
''n''''e''
''n'')
in variables ''x''
1, ..., ''x''
''n''.
In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
norm forms are studied as
Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
s, where they generalize, for example, the
Pell equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinate ...
.
[.] For this application the field ''K'' is usually the rational number field, the field ''L'' is an
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
, and the basis is taken of some
order in the
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
''O''
''L'' of ''L''.
See also
*
Trace form
In mathematics, the field trace is a particular function defined with respect to a finite field extension ''L''/''K'', which is a ''K''-linear map from ''L'' onto ''K''.
Definition
Let ''K'' be a field and ''L'' a finite extension (and hence an ...
References
{{reflist
Field (mathematics)
Diophantine equations
Homogeneous polynomials