Kummer quadratic reciprocity numbers valuation algebra ideals number fields algebraic number fields integers
The grounds of the subject as we know it were set in the late nineteenth century, when ideal numbers, the theory of ideals and valuation theory were developed; these are three complementary ways of dealing with the lack of unique factorisation in algebraic number fields. (For example, in the field generated by the rationals
and 





−
5




{\displaystyle {\sqrt {-5}}}

, the number 



6


{\displaystyle 6}

 can be factorised both as 



6
=
2
⋅
3


{\displaystyle 6=2\cdot 3}

 and




6
=
(
1
+


−
5


)
(
1
−


−
5


)


{\displaystyle 6=(1+{\sqrt {-5}})(1-{\sqrt {-5}})}

; all of 



2


{\displaystyle 2}

, 



3


{\displaystyle 3}

, 



1
+


−
5




{\displaystyle 1+{\sqrt {-5}}}

 and




1
−


−
5




{\displaystyle 1-{\sqrt {-5}}}


are irreducible, and thus, in a naïve sense, analogous to primes among the integers.) The initial impetus for the development of ideal numbers (by Kummer) seems to have come from the study of higher reciprocity laws, that is, generalisations of quadratic reciprocity.
