Quadratic field
From Wikipedia, the free encyclopedia
In mathematics, a quadratic field is an algebraic number field K of degree two over Q. It is easy to show that the map d↦Q(√d) is a bijection from the set of all square-free integers d 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 its archimedean embeddings are real or complex.
Quadratic fields are a basic object of study and class of examples in algebraic number theory. They have been studied in great depth, initially as part of the theory of quadratic forms. There remain some unsolved problems. The class number problem is particularly important.
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[edit] Discriminant
The discriminant of the quadratic field Q(√d) 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 by 1 and the square root of d only in the second case, and in the first case there are such integers that lie at half the 'lattice points' (for example, when d = −3, these are the Eisenstein integers, given by the complex cube roots of unity).
[edit] 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.
If one takes the other cyclotomic fields, they have Galois groups with extra 2-torsion, and so contain at least three quadratic fields.
[edit] Prime factorization into ideals
Any prime number p gives rise to an ideal pOK in the ring of integers OK of a quadratic field K. In line with general theory of splitting of prime ideals in Galois extensions, this may be
- 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 only for the primes dividing the discriminant. The other two cases both occur, as p varies, and in a certain sense are equally likely.
