1911 Encyclopædia Britannica/Number/Quadratic Fields

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 $m=23$.