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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 ...
The minimal polynomial over K of θ is thus the monic polynomial of minimal degree that has θ as a root. Because L is a field, this minimal polynomial is necessarily irreducible over K. For example, the minimal polynomial (over the reals as well as over the rationals) of the complex number i is +. The cyclotomic polynomials are the minimal ...
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 ...
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.
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.
This polynomial is called its minimal polynomial. If its minimal polynomial has degree n, then the algebraic number is said to be of degree n. For example, all rational numbers have degree 1, and an algebraic number of degree 2 is a quadratic irrational. The algebraic numbers are dense in the reals. This follows from the fact they contain the ...
A minimal algebra is a finite algebra with more than one element, in which every non-constant unary polynomial is a permutation on its domain. In simpler terms, it’s an algebraic structure where unary operations (those involving a single input) behave like permutations (bijective mappings). These algebras provide intriguing connections ...
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