In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q, the rational numbers. The map d ↦ Q(√) is a bijection from the set of all square-free integers d ≠ 0, 1 to the set of all quadratic fields. If d > 0 the corresponding quadratic field is called a real quadratic field, and for d < 0 an imaginary quadratic field or complex quadratic field, corresponding to whether it is or not a subfield of the field of the real numbers.
Ring of integers
For a nonzero square free integer d, the discriminant of the quadratic field K=Q(√) is d if d is congruent to 1 modulo 4, and otherwise 4d. For example, when d is −1 so that K is the field of so-called Gaussian rationals, the discriminant is −4. The reason for this distinction relates to general algebraic number theory. The ring of integers of K is spanned over the rational integers by 1 and √ only in the second case, while in the first case it is spanned by 1 and (1 + √)/2.
The set of discriminants of quadratic fields is exactly the set of fundamental discriminants.
Prime factorization into ideals
- p is inert
- (p) is a prime ideal
- The quotient ring is the finite field with p2 elements: OK/pOK = Fp2
- p splits
- (p) is a product of two distinct prime ideals of OK.
- The quotient ring is the product OK/pOK = Fp × Fp.
- p is ramified
- (p) is the square of a prime ideal of OK.
- The quotient ring contains non-zero nilpotent elements.
The third case happens if and only if p divides the discriminant D. The first and second cases occur when the Kronecker symbol (D/p) equals −1 and +1, respectively. For example, if p is an odd prime not dividing D, then p splits if and only if D is congruent to a square modulo p. The first two cases are in a certain sense equally likely to occur as p runs through the primes, see Chebotarev density theorem.
The law of quadratic reciprocity implies that the splitting behaviour of a prime p in a quadratic field depends only on p modulo D, where D is the field discriminant.
Quadratic subfields of cyclotomic fields
The quadratic subfield of the prime cyclotomic field
A classical example of the construction of a quadratic field is to take the unique quadratic field inside the cyclotomic field generated by a primitive p-th root of unity, with p a prime number > 2. The uniqueness is a consequence of Galois theory, there being a unique subgroup of index 2 in the Galois group over Q. As explained at Gaussian period, the discriminant of the quadratic field is p for p = 4n + 1 and −p for p = 4n + 3. This can also be predicted from enough ramification theory. In fact p is the only prime that ramifies in the cyclotomic field, so that p is the only prime that can divide the quadratic field discriminant. That rules out the 'other' discriminants −4p and 4p in the respective cases.
Other cyclotomic fields
If one takes the other cyclotomic fields, they have Galois groups with extra 2-torsion, and so contain at least three quadratic fields. In general a quadratic field of field discriminant D can be obtained as a subfield of a cyclotomic field of D-th roots of unity. This expresses the fact that the conductor of a quadratic field is the absolute value of its discriminant, a special case of the Führerdiskriminantenproduktformel.
- Class number problem
- Eisenstein–Kronecker number
- Heegner number
- Infrastructure (number theory)
- Quadratic integer
- Quadratic irrational
- Stark–Heegner theorem
- Samuel, pp. 76–77
- Buell, Duncan (1989). Binary quadratic forms: classical theory and modern computations. Springer-Verlag. ISBN 0-387-97037-1. Chapter 6.
- Samuel, Pierre (1972). Algebraic number theory. Hermann/Kershaw.
- Stewart, I. N.; Tall, D. O. (1979). Algebraic number theory. Chapman and Hall. ISBN 0-412-13840-9. Chapter 3.1.