Editing Quadratic field (section)

==Prime factorization into ideals==
Any prime number <math>p</math> gives rise to an ideal <math>p\mathcal{O}_K</math> in the [[ring of integers]] <math>\mathcal{O}_K</math> of a quadratic field <math>K</math>. In line with general theory of [[splitting of prime ideals in Galois extensions]], this may be<ref name=":0">{{Cite web|title=Number Rings|url=http://websites.math.leidenuniv.nl/algebra/ant.pdf|last=Stevenhagen|pages=36}}</ref>

;<math>p</math> is '''inert''': <math>(p)</math> is a prime ideal.
: The quotient ring is the [[finite field]] with <math>p^2</math> elements: <math>\mathcal{O}_K / p\mathcal{O}_K = \mathbf{F}_{p^2}</math>.
;<math>p</math> '''splits''': <math>(p)</math> is a product of two distinct prime ideals of <math>\mathcal{O}_K</math>.
: The quotient ring is the product <math>\mathcal{O}_K/p\mathcal{O}_K = \mathbf{F}_p\times\mathbf{F}_p</math>.
;<math>p</math> is '''ramified''': <math>(p)</math> is the square of a prime ideal of <math>\mathcal{O}_K</math>.
:The quotient ring contains non-zero [[nilpotent]] elements.

The third case happens if and only if <math>p</math> divides the discriminant <math>D</math>.  The first and second cases occur when the [[Kronecker symbol]] <math>(D/p)</math> equals <math>-1</math> and <math>+1</math>, respectively. For example, if <math>p</math> is an odd prime not dividing <math>D</math>, then <math>p</math> splits if and only if <math>D</math> is congruent to a square modulo <math>p</math>.  The first two cases are, in a certain sense, equally likely to occur as <math>p</math> runs through the primes—see [[Chebotarev density theorem]].<ref>{{Harvnb|Samuel|1972|pp=76f}}</ref>

The law of [[quadratic reciprocity]] implies that the splitting behaviour of a prime <math>p</math> in a quadratic field depends only on <math>p</math> modulo <math>D</math>, where <math>D</math> is the field discriminant.