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
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 .