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In general a quadratic field of field discriminant can be obtained as a subfield of a cyclotomic field of -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 conductor-discriminant formula.
An integral basis is given by {1, α, α(α + 1)/2} and the discriminant of K is −503. [5] [6] Repeated discriminants: the discriminant of a quadratic field uniquely identifies it, but this is not true, in general, for higher-degree number fields. For example, there are two non-isomorphic cubic fields of discriminant 3969.
A quadratic field is a field extension of the rational numbers that has degree 2. The discriminant of a quadratic field plays a role analogous to the discriminant of a quadratic form. There exists a fundamental connection: an integer is a fundamental discriminant if and only if:
On the other hand, the maximal real subfields Q(cos(2π/2 n)) of the 2-power cyclotomic fields Q(ζ 2 n) (where n is a positive integer) are known to have class number 1 for n≤8, [8] and it is conjectured that they have class number 1 for all n. Weber showed that these fields have odd class number.
By definition, the different ideal δ K is the inverse fractional ideal I −1: it is an ideal of O K. The ideal norm of δ K is equal to the ideal of Z generated by the field discriminant D K of K. The different of an element α of K with minimal polynomial f is defined to be δ(α) = f′(α) if α generates the field K (and zero otherwise ...
Let D be the discriminant of the field, n be the degree of K over , and = be the number of complex embeddings where is the number of real embeddings.Then every class in the ideal class group of K contains an integral ideal of norm not exceeding Minkowski's bound
The conductor of N over k is f, and f 2 is the relative discriminant of N over K. The discriminant of N is d 3 f 4. [6] [7] The field K is a pure cubic field if and only if d = −3. This is the case for which the quadratic field contained in the Galois closure of K is the cyclotomic field of cube roots of unity. [7]
For given low class number (such as 1, 2, and 3), Gauss gives lists of imaginary quadratic fields with the given class number and believes them to be complete. Infinitely many real quadratic fields with class number one Gauss conjectures that there are infinitely many real quadratic fields with class number one.