# Norm form

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*.[1] 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 norm forms are studied as Diophantine equations, where they generalize, for example, the Pell equation.[2] 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 *O*_{L} of *L*.

## References

- Lekkerkerker, Cornelis Gerrit (1969),
*Geometry of numbers*, Bibliotheca Mathematica,**8**, Amsterdam: North-Holland Publishing Co., p. 29, MR 0271032. - Bombieri, Enrico; Gubler, Walter (2006),
*Heights in Diophantine geometry*, New Mathematical Monographs,**4**, Cambridge University Press, Cambridge, pp. 190–191, doi:10.1017/CBO9780511542879, ISBN 978-0-521-84615-8, MR 2216774.

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