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In linear algebra, the minimal polynomial μ A of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. Any other polynomial Q with Q(A) = 0 is a (polynomial) multiple of μ A. The following three statements are equivalent: λ is a root of μ A, λ is a root of the characteristic polynomial χ A ...
Minimal polynomials are useful for constructing and analyzing field extensions. When α is algebraic with minimal polynomial f(x), the smallest field that contains both F and α is isomorphic to the quotient ring F[x]/ f(x) , where f(x) is the ideal of F[x] generated by f(x). Minimal polynomials are also used to define conjugate elements.
Finding roots of polynomials has been an important problem since the time of the ancient Greeks. Some polynomials, however, such as x 2 + 1 over R, the real numbers, have no roots. By constructing the splitting field for such a polynomial one can find the roots of the polynomial in the new field.
An algebraic extension is a purely inseparable extension if and only if for every , the minimal polynomial of over F is not a separable polynomial. [1] If F is any field, the trivial extension is purely inseparable; for the field F to possess a non-trivial purely inseparable extension, it must be imperfect as outlined in the above section.
In mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element α, over a field extension L/K, are the roots of the minimal polynomial p K,α (x) of α over K. Conjugate elements are commonly called conjugates in contexts where this is not ambiguous.
A polynomial P is annihilating or called an annihilating polynomial in linear algebra and operator theory if the polynomial considered as a function of the linear operator or a matrix A evaluates to zero, i.e., is such that P(A) = 0. Note that all characteristic polynomials and minimal polynomials of A are annihilating polynomials
For instance, the polynomial g(X) = X 2 − 1 has precisely deg g = 2 roots in the complex plane; namely 1 and −1, and hence does have distinct roots. On the other hand, the polynomial h(X) = (X − 2) 2, which is the square of a non-constant polynomial does not have distinct roots, as its degree is two, and 2 is its only root.
In number theory, the real parts of the roots of unity are related to one-another by means of the minimal polynomial of (/). The roots of the minimal polynomial are twice the real part of the roots of unity, where the real part of a root of unity is just cos ( 2 k π / n ) {\displaystyle \cos \left(2k\pi /n\right)} with k {\displaystyle ...
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