In algebraic number theory, a quadratic field is an algebraic number field of degree two over Q {\displaystyle \mathbf {Q} } , the rational numbers. Every such quadratic field is some Q ( d ) {\displaystyle \mathbf {Q} ({\sqrt {d}})} where d {\displaystyle d} is a (uniquely defined) square-free integer different from 0 {\displaystyle 0} and 1 {\displaystyle 1} . If d > 0 {\displaystyle d>0} , the corresponding quadratic field is called a real quadratic field, and, if d < 0 {\displaystyle d<0} , it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a subfield of the field of the real numbers. Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms. There remain some unsolved problems. The class number problem is particularly important.
Ring of integers
Discriminant For a nonzero square free integer d {\displaystyle d} , the discriminant of the quadratic field K = Q ( d ) {\displaystyle K=\mathbf {Q} ({\sqrt {d}})} is d {\displaystyle d} if d {\displaystyle d} is congruent to 1 {\displaystyle 1} modulo 4 {\displaystyle 4} , and otherwise 4 d {\displaystyle 4d} . For example, if d {\displaystyle d} is − 1 {\displaystyle -1} , then K {\displaystyle K} is the field of Gaussian rationals and the discriminant is − 4 {\displaystyle -4} . The reason for such a distinction is that the ring of integers of K {\displaystyle K} is generated by ( 1 + d ) / 2 {\displaystyle (1+{\sqrt {d}})/2} in the first case and by d {\displaystyle {\sqrt {d}}} in the second case. The set of discriminants of quadratic fields is exactly the set of fundamental discriminants (apart from 1 {\displaystyle 1} , which is a fundamental discriminant but not the discriminant of a quadratic field).
Prime factorization into ideals Any prime number p {\displaystyle p} gives rise to an ideal p O K {\displaystyle p{\mathcal {O}}_{K}} in the ring of integers O K {\displaystyle {\mathcal {O}}_{K}} of a quadratic field K {\displaystyle K} . In line with general theory of splitting of prime ideals in Galois extensions, this may be
p {\displaystyle p} is inert
( p ) {\displaystyle (p)} is a prime ideal. The quotient ring is the finite field with p 2 {\displaystyle p^{2}} elements: O K / p O K = F p 2 {\displaystyle {\mathcal {O}}_{K}/p{\mathcal {O}}_{K}=\mathbf {F} _{p^{2}}} .
p {\displaystyle p} splits
( p ) {\displaystyle (p)} is a product of two distinct prime ideals of O K {\displaystyle {\mathcal {O}}_{K}} . The quotient ring is the product O K / p O K = F p × F p {\displaystyle {\mathcal {O}}_{K}/p{\mathcal {O}}_{K}=\mathbf {F} _{p}\times \mathbf {F} _{p}} .
p {\displaystyle p} is ramified
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