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

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 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 OL of L.


  1. Lekkerkerker, Cornelis Gerrit (1969), Geometry of numbers, Bibliotheca Mathematica, 8, Amsterdam: North-Holland Publishing Co., p. 29, MR 0271032.
  2. 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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