What is the definition of a degree monomial?

In mathematics, particularly in algebra and algebraic geometry, a degree monomial is a term in a polynomial expression that consists of a single variable raised to a non-negative integer power, multiplied by a non-zero constant coefficient. The degree of a monomial is the sum of the exponents of its variables.

Here is the formal definition:

Let \(R\) be a commutative ring with unity. Let \(x\) be an indeterminate and \(d \in \mathbb{N}_0\). Then, a degree monomial is an element of \(R\) of the form \(c x^d\), where \(c \in R\) is the coefficient of the monomial and \(d\) is the degree of the monomial.

For example, in the polynomial \(2x^3 - 3x^2 + 5x - 1\), the following are degree monomials:

- \(2x^3\), with degree 3

- \(-3x^2\), with degree 2

- \(5x\), with degree 1

The monomial \(-1\) is also a degree monomial, but it has degree 0.

Degree monomials play an important role in various areas of mathematics, including algebraic geometry, commutative algebra, and algebraic number theory. They are often used as building blocks to construct more complex polynomial expressions and equations.