Search results
Results from the WOW.Com Content Network
If the discriminant is positive, the number of non-real roots is a multiple of 4. That is, there is a nonnegative integer k ≤ n/4 such that there are 2k pairs of complex conjugate roots and n − 4k real roots. If the discriminant is negative, the number of non-real roots is not a multiple of 4.
Richard Dedekind showed that every number field possesses an integral basis, allowing him to define the discriminant of an arbitrary number field. [16] 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 ...
More generally, for any square-free integer , the quadratic field is a number field obtained by adjoining the square root of to the field of rational numbers. Arithmetic operations in this field are defined in analogy with the case of Gaussian rational numbers, d = − 1 {\displaystyle d=-1} .
K is a number field. [K : Q] = n = r 1 + 2r 2, where r 1 denotes the number of real embeddings of K, and 2r 2 is the number of complex embeddings of K. ζ K (s) is the Dedekind zeta function of K. h K is the class number, the number of elements in the ideal class group of K. Reg K is the regulator of K. w K is the number of roots of unity ...
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.
If the discriminant is positive, then there are two distinct roots +, both of which are real numbers. For quadratic equations with rational coefficients, if the discriminant is a square number , then the roots are rational—in other cases they may be quadratic irrationals .
Since the number of integral ideals of given norm is finite, the finiteness of the class number is an immediate consequence, [1] and further, the ideal class group is generated by the prime ideals of norm at most M K. Minkowski's bound may be used to derive a lower bound for the discriminant of a field K given n, r 1 and r 2.
Typically, R is the ring of the integers, the field of fractions is the field of the rational numbers and the algebraically closed field is the field of the complex numbers. Vieta's formulas are then useful because they provide relations between the roots without having to compute them.