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...

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...

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...

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...

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

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

\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...

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

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...

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...

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...

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...

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 + ⋯...

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

of several algorithms, including polynomial long division and synthetic division. When the denominator b(x) is monic and linear, that is, b(x) = x − c...

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

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...

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

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...

\\\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...

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...

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...

is said to be integral over a subring A of B if b is a root of some monic polynomial over A. If A, B are fields, then the notions of "integral over" and...

contained in K {\displaystyle K} . An algebraic integer is a root of a monic polynomial with integer coefficients: x n + c n − 1 x n − 1 + ⋯ + c 0 {\displaystyle...