If *D*_{R} is odd and in the above notation *N* is even, then the primes
lying over 2 and
^{ . }2 + ^{ . } and
^{ . }2 + ^{ . }(1 - ). When *N* is odd, then the only prime lying
over 2 is 2*R*. Now, if *p* is an odd prime that does not divide the
discriminant then either
_{p}[*T*]/(*P*(*T*)) is isomorphic to
_{p2} or it splits into two
_{p} factors. The former case
occurs when *D*_{R} is not a square modulo *p* and in this case the
prime lying over *p* is just the ideal *pR*; which is principal. In
the second case *D*_{R} is a square modulo *p* and we obtain two primes
*P*_{p} and *Q*_{p}, lying over *p*; both these have norm *p* and their
product (and intersection) is *pR*. Let *c*_{p} be a number between 1
and *p* - 1 so that
*c*_{p}^{2} = *D*_{R}(mod *p*); then
*a*_{p} = (1 + *c*_{p})/2 satisfies
the equation modulo *p* in the *D*_{R} odd case and
*a*_{p} = (*c*_{p})/2
satisfies the equation modulo *p* in the *D*_{R} even case. Thus we can
pick a solution *a*_{p} of the equation modulo *p* in each case and
declare that
*P*_{p} = ^{ . }*p* + ^{ . }( - *a*_{p}). The primes *P*_{p}
and *Q*_{p} are interchanged by the involution.