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
- e1, ..., en
for L as a vector space over K, the form is given by
- N(x1e1 + ... + xnen)
- x1, ..., xn.
In number theory norm forms are studied as Diophantine equations, 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.
- 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.