1911 Encyclopædia Britannica/Number/Quadratic Fields

52. Quadratic Fields.&mdash;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 &sect;&sect; 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 &sect; 44 for the case $$m = 23$$.