1911 Encyclopædia Britannica/Number/Quadratic Fields

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52. Quadratic Fields.—Let m be an ordinary integer different from +1, and not divisible by any square: then if x, y assume all ordinary rational values the expressions x+y\surd m are the elements of a field which may be called \Omega(\surd m). It should be observed that \surd m means one definite root of x^2-m=0, it does not matter which: it is convenient, however, to agree that \surd m is positive when m is positive, and i\surd m is negative when m is negative. The principal results relating to \Omega will now be stated, and will serve as illustrations of §§ 44–51.

In the notation previously used

\mathfrak{o} = [1, \tfrac{1}{2}(1+\surd m)] \mbox{ or } [1, \surd m]

according as m \equiv 1 \pmod{4} or not. In the first case \Delta = m, in the second \Delta = 4m. The field \Omega is normal, and every ideal prime in it is of the first degree.

Let q be any odd prime factor of m; then q = \mathfrak{q}^2, where \mathfrak{q} is the prime ideal [q, \tfrac{1}{2}(q+\surd m)] when m \equiv 1 \pmod{4} and in other cases [q, \surd m]. An odd prime p of which m is a quadratic residue is the product of two prime ideals \mathfrak{p}, \mathfrak{p}', which may be written in the form [p, \tfrac{1}{2}(a+\surd m)], [p, \tfrac{1}{2}(a-\surd m)] or [p, a+\surd m], [p, a-\surd m], according as m \equiv 1 \pmod{4} or not: here a is a root of x^2 \equiv m \pmod{p}, taken so as to be odd in the first of the two cases. All other rational odd primes are primes in \Omega. For the exceptional prime 2 there are four cases to consider: (i.) If m \equiv 1 \pmod{8}, then 2 = [2, \tfrac{1}{2}(1+\surd m)] \times [2, \tfrac{1}{2}(1-\surd m)]. ( ii.) If m \equiv 5 \pmod{8}, then 2 is prime: (iii.) if m \equiv 2 \pmod{4}, 2 = [2, \surd m]^2: (iv.) if m \equiv 3 \pmod{4}, 2 = [2, 1+\surd m]^2. Illustrations will be found in § 44 for the case m = 23.