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  2. Discriminant - Wikipedia

    en.wikipedia.org/wiki/Discriminant

    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.

  3. Discriminant of an algebraic number field - Wikipedia

    en.wikipedia.org/wiki/Discriminant_of_an...

    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 ...

  4. Resolvent (Galois theory) - Wikipedia

    en.wikipedia.org/wiki/Resolvent_(Galois_theory)

    The simplest examples of resolvents are X 2 − Δ {\displaystyle X^{2}-\Delta } where Δ {\displaystyle \Delta } is the discriminant , which is a resolvent for the alternating group . In the case of a cubic equation , this resolvent is sometimes called the quadratic resolvent ; its roots appear explicitly in the formulas for the roots of a ...

  5. Finite field - Wikipedia

    en.wikipedia.org/wiki/Finite_field

    The field GF(q) contains a n th primitive root of unity if and only if n is a divisor of q − 1; if n is a divisor of q − 1, then the number of primitive n th roots of unity in GF(q) is φ(n) (Euler's totient function). The number of n th roots of unity in GF(q) is gcd(n, q − 1).

  6. Casus irreducibilis - Wikipedia

    en.wikipedia.org/wiki/Casus_irreducibilis

    Casus irreducibilis occurs when none of the roots are rational and when all three roots are distinct and real; the case of three distinct real roots occurs if and only if ⁠ q 2 / 4 ⁠ + ⁠ p 3 / 27 ⁠ < 0, in which case Cardano's formula involves first taking the square root of a negative number, which is imaginary, and then taking the ...

  7. Quadratic equation - Wikipedia

    en.wikipedia.org/wiki/Quadratic_equation

    Figure 1. Plots of quadratic function y = ax 2 + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0). A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots.

  8. Vieta's formulas - Wikipedia

    en.wikipedia.org/wiki/Vieta's_formulas

    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.

  9. Class number formula - Wikipedia

    en.wikipedia.org/wiki/Class_number_formula

    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 ...