52. Quadratic Fields.—Let
be an ordinary integer different from
, and not divisible by any square: then if
assume all ordinary rational values the expressions
are the elements of a field which may be called
. It should be observed that
means one definite root of
, it does not matter which: it is convenient, however, to agree that
is positive when
is positive, and
is negative when
is negative. The principal results relating to
will now be stated, and will serve as illustrations of §§ 44–51.
In the notation previously used
![{\displaystyle {\mathfrak {o}}=[1,{\tfrac {1}{2}}(1+\surd m)]{\mbox{ or }}[1,\surd m]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49d354abc369b599822dd75a68168d123343ac69)
according as
or not. In the first case
, in the second
. The field
is normal, and every ideal prime in it is of the first degree.
Let
be any odd prime factor of
; then
, where
is the prime ideal
when
and in other cases
. An odd prime
of which
is a quadratic residue is the product of two prime ideals
, which may be written in the form
or
, according as
or not: here
is a root of
, taken so as to be odd in the first of the two cases. All other rational odd primes are primes in
. For the exceptional prime
there are four cases to consider: (i.) If
, then
. ( ii.) If
, then
is prime: (iii.) if
: (iv.) if
. Illustrations will be found in § 44 for the case
.