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.

## See also

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* Trace form

## References

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1. ^ Lekkerkerker, Cornelis Gerrit (1969), Geometry of numbers, Bibliotheca Mathematica, vol. 8, Amsterdam: North-Holland Publishing Co., p. 29, ISBN 9781483259277, MR 0271032.
2. ^ Bombieri, Enrico; Gubler, Walter (2006), Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, pp. 190–191, doi:10.1017/CBO9780511542879, ISBN 978-0-521-84615-8, MR 2216774.