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 nonzero coefficient of highest degree) is equal to 1.
Monic polynomials are widely used in algebra and number theory, since they produce many simplifications and they avoid divisions and denominators.
Vieta's formulas are simpler in the case of monic polynomials: The ith elementary symmetric function of the roots of a monic polynomial of degree n equals
Therefore, it is defined for polynomials with coefficients in a commutative ring.
Divisibility induces a partial order on monic polynomials.
For example, the equation is equivalent to the monic equation When the coefficients are unspecified, or belong to a field where division does not result into fractions (such as
or a finite field), this reduction to monic equations may provide simplification.
On the other hand, as shown by the previous example, when the coefficients are explicit integers, the associated monic polynomial is generally more complicated.
Monic polynomial equations are at the basis of the theory of algebraic integers, and, more generally of integral elements.
Let R be a subring of a field F; this implies that R is an integral domain.
An element a of F is integral over R if it is a root of a monic polynomial with coefficients in R. A complex number that is integral over the integers is called an algebraic integer.
Conversely, an integer p is a root of the monic polynomial
These concepts are fundamental in algebraic number theory.
For example, many of the numerous wrong proofs of the Fermat's Last Theorem that have been written during more than three centuries were wrong because the authors supposed wrongly that the algebraic integers in an algebraic number field have unique factorization.
Ordinarily, the term monic is not employed for polynomials of several variables.
Being monic depends thus on the choice of one "main" variable.
In this case, a polynomial may be said to be monic, if it has 1 as its leading coefficient (for the monomial order).