mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and...

a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root. The complex conjugate of a complex number...

coefficients (this is the complex conjugate root theorem). Conversely, if one has a factor of degree two, the quadratic formula gives a root. Every univariate...

[L:K(α)]sep. A theorem of Kronecker states that if α is a nonzero algebraic integer such that α and all of its conjugates in the complex numbers have absolute...

square roots Cube root Root of unity Constructible number Complex conjugate root theorem Algebraic element Horner scheme Rational root theorem Gauss's lemma...

exponents. In particular, the reciprocal of an nth root of unity is its complex conjugate, and is also an nth root of unity: 1 z = z − 1 = z n − 1 = z ¯ . {\displaystyle...

vanishes. If E is a complex vector bundle, then the conjugate bundle E ¯ {\displaystyle {\overline {E}}} of E is obtained by having complex numbers acting...

the coefficients can be ill-conditioned for many inputs. The complex conjugate root theorem states that if the coefficients of a polynomial are real, then...

electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is looked for. Three non-collinear...

are precisely the locally invertible complex analytic functions. In three and higher dimensions, Liouville's theorem sharply limits the conformal mappings...

element) of order p in G. Theorem (2) — Given a finite group G and a prime number p, all Sylow p-subgroups of G are conjugate to each other. That is, if...

Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions f and g holomorphic inside some region K {\displaystyle...

In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function...

domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also...

In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f {\displaystyle f} , is a member x {\displaystyle...

scalars is the complex numbers. By the fundamental theorem of algebra, applied to the characteristic polynomial of A, there is at least one complex eigenvalue...

contour integral that is not necessarily zero, by the residue theorem. Cutting the complex plane ensures not only that Γ(z) is holomorphic in this restricted...

real coefficients, then its complex conjugate is also a root. So the non-real roots, if any, occur as pairs of complex conjugate roots. As a cubic polynomial...

In complex analysis, the Riemann mapping theorem states that if U {\displaystyle U} is a non-empty simply connected open subset of the complex number...

including points at infinity and points with complex coordinates. In its modern formulation, the theorem states that, if N is the number of common points...

{x} ~.} More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number   z ∗...

fundamental theorem of algebra, a polynomial of degree n has exactly n real or complex roots counting multiplicities. It follows that the problem of root finding...

simple root of the characteristic polynomial. (These requirements hold for primitive matrices as above). Given that M is diagonalizable, M is conjugate to...

1} . A split-complex number has two real number components x and y, and is written z = x + y j . {\displaystyle z=x+yj.} The conjugate of z is z ∗ =...

factorization. In fact, if r = a + ib is a non-real root of P(x), then its complex conjugate s = a - ib is also a root of P(x). So, the product ( x − r ) ( x − s...

In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometric relation between the roots of a polynomial P and the roots of its...

solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included and a double root is counted for...

Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law...

plane from the origin. This can be computed using the Pythagorean theorem: for any complex number z = x + i y , {\displaystyle z=x+iy,} where x {\displaystyle...

the primitive element theorem to write K = Q ( α ) {\displaystyle K=\mathbb {Q} (\alpha )} , and then r1 is the number of conjugates of α that are real,...