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.
$\mathfrak{o} = [1, \tfrac{1}{2}(1+\surd m)] \mbox{ or } [1, \surd m]$
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 $m = 23$.