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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 integer that occurs as the discriminant of a quadratic number field is called a fundamental discriminant. [3] Cyclotomic fields: let n > 2 be an integer, let ζ n be a primitive nth root of unity, and let K n = Q(ζ n) be the nth cyclotomic field. The discriminant of K n is given by [2] [4]
Geometrically, the discriminant of a quadratic form in three variables is the equation of a quadratic projective curve. The discriminant is zero if and only if the curve is decomposed in lines (possibly over an algebraically closed extension of the field). A quadratic form in four variables is the equation of a projective surface.
Simultaneously generalizing the case of imaginary quadratic fields and cyclotomic fields is the case of a CM field K, i.e. a totally imaginary quadratic extension of a totally real field. In 1974, Harold Stark conjectured that there are finitely many CM fields of class number 1. [12] He showed that there are finitely many of a fixed degree.
Its discriminant as quadratic form need not be +1 (in fact this happens only for the case K = Q). Define the inverse different or codifferent [ 3 ] [ 4 ] or Dedekind's complementary module [ 5 ] as the set I of x ∈ K such that tr( xy ) is an integer for all y in O K , then I is a fractional ideal of K containing O K .
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Fix a field k of characteristic not equal to 2. All vector spaces will be assumed to be finite-dimensional.Two spaces equipped with symmetric bilinear forms are equivalent if one can be obtained from the other by adding a metabolic quadratic space, that is, zero or more copies of a hyperbolic plane, the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector. [1]
For the real quadratic field = (with d square-free), the fundamental unit ε is commonly normalized so that ε > 1 (as a real number). Then it is uniquely characterized as the minimal unit among those that are greater than 1. If Δ denotes the discriminant of K, then the fundamental unit is