# Hilbert's eleventh problem

Hilbert's eleventh problem is one of David Hilbert's list of open mathematical problems posed at the Second International Congress of Mathematicians in Paris in 1900. A furthering of the theory of quadratic forms, he stated the problem as follows:

Our present knowledge of the theory of quadratic number fields puts us in a position to attack successfully the theory of quadratic forms with any number of variables and with any algebraic numerical coefficients. This leads in particular to the interesting problem: to solve a given quadratic equation with algebraic numerical coefficients in any number of variables by integral or fractional numbers belonging to the algebraic realm of rationality determined by the coefficients.[1]

As stated by Kaplansky, "The 11th Problem is simply this: classify quadratic forms over algebraic number fields." This is exactly what Minkowski did for quadratic form with fractional coefficients. A quadratic form (not quadratic equation) is any polynomial in which each term has variables appearing exactly twice. The general form of such an equation is ax^(2)+bxy+cy^(2).(All coefficients must be whole numbers).

``` 2. Yandell, Benjamin H. The Honors Class: Hilbert's Problems and Their Solvers. Natik: K Peters. Print.