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Let () be a polynomial equation, where P is a univariate polynomial of degree n.If one divides all coefficients of P by its leading coefficient, one obtains a new polynomial equation that has the same solutions and consists to equate to zero a monic polynomial.
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 f of α is unique.. To prove this, suppose that f and g are monic polynomials in J α of minimal degree n > 0. We have that r := f−g ∈ J α (because the latter is closed under addition/subtraction) and that m := deg(r) < n (because the polynomials are monic of the same degree).
That polynomial differs from the one defined here by a sign (), so it makes no difference for properties like having as roots the eigenvalues of ; however the definition above always gives a monic polynomial, whereas the alternative definition is monic only when is even.
The roots of the characteristic polynomial () are the eigenvalues of ().If there are n distinct eigenvalues , …,, then () is diagonalizable as () =, where D is the diagonal matrix and V is the Vandermonde matrix corresponding to the λ 's: = [], = [].
Irreducible polynomials over finite fields are also useful for pseudorandom number generators using feedback shift registers and discrete logarithm over F 2 n. The number of irreducible monic polynomials of degree n over F q is the number of aperiodic necklaces, given by Moreau's necklace-counting function M q (n).
Applied to the monic polynomial + = with all coefficients a k considered as free parameters, this means that every symmetric polynomial expression S(x 1,...,x n) in its roots can be expressed instead as a polynomial expression P(a 1,...,a n) in terms of its coefficients only, in other words without requiring knowledge of the roots.
In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field GF(p m).This means that a polynomial F(X) of degree m with coefficients in GF(p) = Z/pZ is a primitive polynomial if it is monic and has a root α in GF(p m) such that {,,,,, …} is the entire field GF(p m).