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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.
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 ...
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 ...
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.
Otherwise, θ is algebraic over K; that is, θ is a root of a polynomial over K. The monic polynomial of minimal degree n, with θ as a root, is called the minimal polynomial of θ. Its degree equals the degree of the field extension, that is, the dimension of L viewed as a K-vector space.
The Conway polynomial C p,n is defined as the lexicographically minimal monic primitive polynomial of degree n over F p that is compatible with C p,m for all m dividing n.This is an inductive definition on n: the base case is C p,1 (x) = x − α where α is the lexicographically minimal primitive element of F p.
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