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The polynomial has a multiple root if and only if its discriminant is zero. 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.
The root discriminant of a degree n number field K is defined by the formula ... For totally real fields, the root discriminant is > 14, with 1229 exceptions.
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 ...
If the characteristic of GF(q) is odd, the discriminant Δ = b 2 − 4ac indicates the number of roots in GF(q) and the classical quadratic formula gives the roots. However, when GF(q) has even characteristic (i.e., q = 2 h for some positive integer h), these formulas are no longer applicable.
Vieta's formulas are frequently used with polynomials with coefficients in any integral domain R.Then, the quotients / belong to the field of fractions of R (and possibly are in R itself if happens to be invertible in R) and the roots are taken in an algebraically closed extension.
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. If the discriminant is zero, then there is exactly one real root − b 2 a , {\displaystyle -{\frac {b}{2a}},} sometimes called a repeated or double root or two equal roots.
Adjoining a root of x 3 + x 2 − 2x − 1 to Q yields a cyclic cubic field, and hence a totally real cubic field. It has the smallest discriminant of all totally real cubic fields, namely 49. [4] The field obtained by adjoining to Q a root of x 3 + x 2 − 3x − 1 is an example of a totally real cubic field that is not cyclic. Its ...
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.