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Concept
Quadratic field
Summary
In algebraic number theory, a quadratic field is an algebraic number field of degree two over , the rational numbers.
Every such quadratic field is some where is a (uniquely defined) square-free integer different from and . If , the corresponding quadratic field is called a real quadratic field, and, if , 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.
Quadratic integer
For a nonzero square free integer , the discriminant of the quadratic field is if is congruent to modulo , and otherwise . For example, if is , then is the field of Gaussian rationals and the discriminant is . The reason for such a distinction is that the ring of integers of is generated by in the first case and by in the second case.
The set of discriminants of quadratic fields is exactly the set of fundamental discriminants.
Any prime number gives rise to an ideal in the ring of integers of a quadratic field . In line with general theory of splitting of prime ideals in Galois extensions, this may be
is inert is a prime ideal.
The quotient ring is the finite field with elements: .
splits is a product of two distinct prime ideals of .
The quotient ring is the product .
is ramified is the square of a prime ideal of .
The quotient ring contains non-zero nilpotent elements.
The third case happens if and only if divides the discriminant . The first and second cases occur when the Kronecker symbol equals and , respectively. For example, if is an odd prime not dividing , then splits if and only if is congruent to a square modulo . The first two cases are, in a certain sense, equally likely to occur as runs through the primes—see Chebotarev density theorem.
The law of quadratic reciprocity implies that the splitting behaviour of a prime in a quadratic field depends only on modulo , where is the field discriminant.
Official source
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