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 e<sub>1</sub>, ..., e<sub>n</sub> for L as a vector space over K, the form is given by
in variables x<sub>1</sub>, ..., x<sub>n</sub>.
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 O<sub>L</sub> of L.