ideals norms chakravala method chakravala numbers rational numbers number fields quadratic forms arithmetic
It could be argued that the simplest kind of number fields (viz., quadratic fields) were already studied by Gauss, as the discussion of quadratic forms in Disquisitiones arithmeticae can be restated in terms of ideals and
norms in quadratic fields. (A quadratic field consists of all
numbers of the form 



a
+
b


d




{\displaystyle a+b{\sqrt {d}}}

, where




a


{\displaystyle a}

 and 



b


{\displaystyle b}

 are rational numbers and 



d


{\displaystyle d}


is a fixed rational number whose square root is not rational.)
For that matter, the 11th-century chakravala method amounts—in modern terms—to an algorithm for finding the units of a real quadratic number field. However, neither Bhāskara nor Gauss knew of number fields as such.
