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, ..., e_n for L as a vector space over K, the form is given by

N(x₁e₁ + ... + x_ne_n)

in variables x₁, ..., 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.

• Trace form