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.{{citation

| last = Lekkerkerker | first = Cornelis Gerrit

| authorlink = Gerrit Lekkerkerker

| location = Amsterdam

| mr = 0271032

| page = 29

| publisher = North-Holland Publishing Co.

| series = Bibliotheca Mathematica

| title = Geometry of numbers

| url = https://books.google.com/books?id=XZ7iBQAAQBAJ&pg=PA29

| volume = 8

| year = 1969| isbn = 9781483259277

}}. 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.{{citation

| last1 = Bombieri | first1 = Enrico

| authorlink1 = Enrico Bombieri

| last2 = Gubler | first2 = Walter

| doi = 10.1017/CBO9780511542879

| isbn = 978-0-521-84615-8

| mr = 2216774

| pages = 190–191

| publisher = Cambridge University Press, Cambridge

| series = New Mathematical Monographs

| title = Heights in Diophantine geometry

| url = https://books.google.com/books?id=3ATnwmGegvsC&pg=PA190

| volume = 4

| year = 2006| url-access = subscription

}}. 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.