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Then the discriminant is the product of the a i, which is well-defined as a class in K/(K ×) 2. 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 ...
The definition of the discriminant of a general algebraic number field, K, was given by Dedekind in 1871. [16] At this point, he already knew the relationship between the discriminant and ramification. [17] Hermite's theorem predates the general definition of the discriminant with Charles Hermite publishing a proof of it in 1857. [18]
For this converse the field discriminant is needed. This is the Dedekind discriminant theorem. In the example above, the discriminant of the number field () with x 3 − x − 1 = 0 is −23, and as we have seen the 23-adic place ramifies. The Dedekind discriminant tells us it is the only ultrametric place that does.
The following table shows some orders of small discriminant of quadratic fields. The maximal order of an algebraic number field is its ring of integers, and the discriminant of the maximal order is the discriminant of the field. The discriminant of a non-maximal order is the product of the discriminant of the corresponding maximal order by the ...
In mathematics, a quadratic function of a single variable is a function of the form [1] = + +,,where is its variable, and , , and are coefficients.The expression + + , especially when treated as an object in itself rather than as a function, is a quadratic polynomial, a polynomial of degree two.
The roots of the quadratic function y = 1 / 2 x 2 − 3x + 5 / 2 are the places where the graph intersects the x-axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.
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The discriminant of a quadratic form, concretely the class of the determinant of a representing matrix in K / (K ×) 2 (up to non-zero squares) can also be defined, and for a real quadratic form is a cruder invariant than signature, taking values of only "positive, zero, or negative".
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