Quadratic Forms, Rings and Genera

Abstract

We have considered quadratic fields $\Phi \left( {\sqrt D } \right)$ of discriminant d. Here we have constructed the ring of integers (see (9.34))

$$ O = \left[ {1,{\beta _d}} \right],{\beta _d} = \left( {d + \sqrt d } \right)/2. $$

(14.1a)

A prime p (> 0 in ℤ) factors into distinct factors, remains inert or ramifies (to a square ideal) according as

$$ \left( {d/p} \right) = + 1, - 1,0. $$

(14.1b)

This process creates factors p|p where p is an ideal in O and N[ p] = p, but it does not tell us if p is principal. Clearly p is principal if and only if the equation ±p = N(π) is solvable for π ∈ O, (for then p = (π)). If we go back to our definition of “principal form” (3.6),

$$ {Q_{D\left( {x,y} \right)}} = \left\{ {\begin{array}{*{20}{c}} {{x^2} - D{y^2}, \left( {d = 4D} \right)} \\ {{x^2} + xy + \left( {1 - D} \right){y^2}/4, \left( {d = D} \right)} \end{array}} \right. $$

(14.2)

we see that the discovery of principal prime ideals is the same as the solvability of ±p = Q_D(x,y).