In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the...

the ideal of polynomials vanishing on α. The minimal polynomial f of α is unique. To prove this, suppose that f and g are monic polynomials in Jα of minimal...

minimal polynomial μA 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...

gives a monic polynomial, whereas the alternative definition is monic only when n {\displaystyle n} is even. To compute the characteristic polynomial of the...

{k}{n}}}\right).} It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers...

non-constant monic polynomial with coefficients in F is irreducible over F, if it is not the product of two non-constant monic polynomials, with coefficients...

integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers...

roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial. Symmetric polynomials also form...

Monic or monic in Wiktionary, the free dictionary. Monic may refer to: Monic morphism, a special kind of morphism in category theory Monic polynomial...

generic polynomial has a different, although related, meaning: a generic polynomial for a finite group G and a field F is a monic polynomial P with coefficients...

to be a −∞. A constant polynomial is either the zero polynomial, or a polynomial of degree zero. A nonzero polynomial is monic if its leading coefficient...

\end{cases}}} For any given n ≥ 1, among the polynomials of degree n with leading coefficient 1 (monic polynomials): f ( x ) = 1 2 n − 1 T n ( x ) {\displaystyle...

alternating polynomials. Given a polynomial, the Vandermonde polynomial of its roots is defined over the splitting field; for a non-monic polynomial, with leading...

the coefficient a, and then complete the square for the resulting monic polynomial. Example: 3 x 2 + 12 x + 27 = 3 [ x 2 + 4 x + 9 ] = 3 [ ( x + 2 ) 2...

minimal polynomial of x. The minimal polynomial of an algebraic element x of L is irreducible, and is the unique monic irreducible polynomial of which...

divide the quadratic equation by ⁠ a {\displaystyle a} ⁠ to obtain a monic polynomial with the same roots. Namely, x 2 + b a x + c a = ( x − α ) ( x − β...

division of polynomials, with less writing and fewer calculations than long division. It is mostly taught for division by linear monic polynomials (known as...

function Nq(n) counts monic polynomials of degree n which are primary (a power of an irreducible); or alternatively irreducible polynomials of all degrees d...

In linear algebra, the Frobenius companion matrix of the monic polynomial p ( x ) = c 0 + c 1 x + ⋯ + c n − 1 x n − 1 + x n {\displaystyle p(x)=c_{0}+c_{1}x+\cdots...

of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable...

as a root of a polynomial with either integer or rational coefficients. Given an algebraic number, there is a unique monic polynomial with rational coefficients...

necessarily have a monic polynomial, so finally multiply this by a constant to make it a monic polynomial. This will be the GCD of the two polynomials as it includes...

determinant of ( λ I n − A ) {\displaystyle (\lambda I_{n}-A)} is a degree-n monic polynomial in λ, so it can be written as p A ( λ ) = λ n + c n − 1 λ n − 1 + ⋯...

easy to visualize. However, a monic polynomial of odd degree must necessarily have a real root. The associated polynomial function in x is continuous,...

Since L n ( α ) ( x ) {\displaystyle L_{n}^{(\alpha )}(x)} is a monic polynomial of degree n {\displaystyle n} in α {\displaystyle \alpha } , there...

\\\end{aligned}}} The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity ∏ j = 1 n ( λ...

mathematics, it is common to require that the greatest common divisor be a monic polynomial. To get this, it suffices to divide every element of the output by...

are precisely qn monic polynomials of degree n (including the reducible ones), this can be rephrased as follows: if a monic polynomial of degree n is selected...

minimal polynomial H(X) of θ over K; it is a monic polynomial with coefficients in OK. Reducing the coefficients of H(X) modulo P, we obtain a monic polynomial...

polynomial of even degree 2d, then there is a polynomial q of degree d such that p(x) = xdq(x + ⁠1/x⁠). If p(x) is a monic antipalindromic polynomial...