52. Quadratic Fields.—Let ${\displaystyle m}$ be an ordinary integer different from ${\displaystyle +1}$, and not divisible by any square: then if ${\displaystyle x,y}$ assume all ordinary rational values the expressions ${\displaystyle x+y\surd m}$ are the elements of a field which may be called ${\displaystyle \Omega (\surd m)}$. It should be observed that ${\displaystyle \surd m}$ means one definite root of ${\displaystyle x^{2}-m=0}$, it does not matter which: it is convenient, however, to agree that ${\displaystyle \surd m}$ is positive when ${\displaystyle m}$ is positive, and ${\displaystyle i\surd m}$ is negative when ${\displaystyle m}$ is negative. The principal results relating to ${\displaystyle \Omega }$ will now be stated, and will serve as illustrations of §§ 44–51.
${\displaystyle {\mathfrak {o}}=[1,{\tfrac {1}{2}}(1+\surd m)]{\mbox{ or }}[1,\surd m]}$
according as ${\displaystyle m\equiv 1{\pmod {4}}}$ or not. In the first case ${\displaystyle \Delta =m}$, in the second ${\displaystyle \Delta =4m}$. The field ${\displaystyle \Omega }$ is normal, and every ideal prime in it is of the first degree.
Let ${\displaystyle q}$ be any odd prime factor of ${\displaystyle m}$; then ${\displaystyle q={\mathfrak {q}}^{2}}$, where ${\displaystyle {\mathfrak {q}}}$ is the prime ideal ${\displaystyle [q,{\tfrac {1}{2}}(q+\surd m)]}$ when ${\displaystyle m\equiv 1{\pmod {4}}}$ and in other cases ${\displaystyle [q,\surd m]}$. An odd prime ${\displaystyle p}$ of which ${\displaystyle m}$ is a quadratic residue is the product of two prime ideals ${\displaystyle {\mathfrak {p}},{\mathfrak {p}}'}$, which may be written in the form ${\displaystyle [p,{\tfrac {1}{2}}(a+\surd m)],[p,{\tfrac {1}{2}}(a-\surd m)]}$ or ${\displaystyle [p,a+\surd m],[p,a-\surd m]}$, according as ${\displaystyle m\equiv 1{\pmod {4}}}$ or not: here ${\displaystyle a}$ is a root of ${\displaystyle x^{2}\equiv m{\pmod {p}}}$, taken so as to be odd in the first of the two cases. All other rational odd primes are primes in ${\displaystyle \Omega }$. For the exceptional prime ${\displaystyle 2}$ there are four cases to consider: (i.) If ${\displaystyle m\equiv 1{\pmod {8}}}$, then ${\displaystyle 2=[2,{\tfrac {1}{2}}(1+\surd m)]\times [2,{\tfrac {1}{2}}(1-\surd m)]}$. ( ii.) If ${\displaystyle m\equiv 5{\pmod {8}}}$, then ${\displaystyle 2}$ is prime: (iii.) if ${\displaystyle m\equiv 2{\pmod {4}},2=[2,\surd m]^{2}}$: (iv.) if ${\displaystyle m\equiv 3{\pmod {4}},2=[2,1+\surd m]^{2}}$. Illustrations will be found in § 44 for the case ${\displaystyle m=23}$.