Monom

In algebra , a monomial is a polynomial that consists of only one term. A monomial is therefore a product consisting of a coefficient and powers of one, and rarely several variables.

Examples of monomials of variables : ${\ displaystyle a}$

{\ displaystyle a, \ a ^ {2}, \ 7b \ cdot a, \ -5b ^ {4} a ^ {2}}

Every polynomial is a sum of monomials of the same variable, for example is

{\ displaystyle \ 5x ^ {3} + 7x ^ {2} -2x-10}

made up of the following monomials:

{\ displaystyle 5x ^ {3}, \ 7x ^ {2}, \ -2x ^ {1}, \ -10x ^ {0}}

Polynomial functions whose function term is a monomial, are power functions .

Alternative definition

In parts of the literature, only the product of the variables (i.e. without coefficients) is referred to as a monomial. If you follow this way of speaking, then the monomials have the following property:

If one considers the polynomial ring in variables over a field as a vector space over , then the set of monomials is a basis of this vector space. ${\ displaystyle K [X_ {1}, \ ldots, X_ {n}]}$ ${\ displaystyle n}$ ${\ displaystyle X_ {1}, \ ldots, X_ {n}}$ ${\ displaystyle K}$ ${\ displaystyle K}$

In the special case of a single variable , this base consists of the monomials ${\ displaystyle X}$

{\ displaystyle 1, \ X, \ X ^ {2}, \ X ^ {3}, \ \ ldots}

generalization

If we allow several variables and arbitrary real powers, we get the monomial functions .

literature

H. Lüneburg: groups, rings, bodies . Oldenbourg, Munich 1999, ISBN 3-486-24977-0 .
Cox, David ; Little, John; O'Shea, Donald: Ideals, varieties, and algorithms . Springer-Verlag, New York 1992, ISBN 0-387-97847-X .