information:highlighting, linguistic:translation
Imaginary quadratic fields

The above laws are special cases of more general laws that hold for the ring of integers in any imaginary quadratic number field. Let k be an imaginary quadratic number field with ring of integers O k. {\displaystyle {\mathcal {O}}_{k}.} For a prime ideal p ⊂ O k {\displaystyle {\mathfrak {p}}\subset {\mathcal {O}}_{k}} with odd norm N p {\displaystyle \mathrm {N} {\mathfrak {p}}} and α ∈ O k, {\displaystyle \alpha \in {\mathcal {O}}_{k},} define the quadratic character for O k {\displaystyle {\mathcal {O}}_{k}} as

for an arbitrary ideal a ⊂ O k {\displaystyle {\mathfrak {a}}\subset {\mathcal {O}}_{k}} factored into prime ideals a = p 1 ⋯ p n {\displaystyle {\mathfrak {a}}={\mathfrak {p}}_{1}\cdots {\mathfrak {p}}_{n}} define

and for β ∈ O k {\displaystyle \beta \in {\mathcal {O}}_{k}} define

Let O k = Z ω 1 ⊕ Z ω 2, {\displaystyle {\mathcal {O}}_{k}=\mathbb {Z} \omega _{1}\oplus \mathbb {Z} \omega _{2},} i.e. { ω 1, ω 2 } {\displaystyle \left\{\omega _{1},\omega _{2}\right\}} is an integral basis for O k. {\displaystyle {\mathcal {O}}_{k}.} For ν ∈ O k {\displaystyle \nu \in {\mathcal {O}}_{k}} with odd norm N ν, {\displaystyle \mathrm {N} \nu,} define (ordinary) integers a, b, c, d by the equations,

and a function

If m = Nμ and n = Nν are both odd, Herglotz proved

Also, if

Then