Indeterminate (variable)

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In mathematics, particularly abstract algebra, an indeterminate is a variable symbol used to describe a class of expressions over a given ring, usually to define a new ring over these expressions. For example, a polynomial over a ring ${\displaystyle R}$, may be defined as an expression of the form:$${\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+...a_{n}X^{n}}$$Where, the coefficients ${\textstyle a_{i},\;0\leq i\leq n,}$ are elements of ${\displaystyle R}$, and ${\displaystyle X}$ is an indeterminate (which is not considered an element of ${\displaystyle R}$). Then, one may define the ring of polynomials over ${\displaystyle R}$ in the indeterminate ${\displaystyle X}$, usually denoted ${\displaystyle R[X]}$. Indeterminates are often used in defining rings over polynomials, algebraic fractions, formal power series, and other algebraic expressions.

A fundamental property of an indeterminate is that it can be substituted with any mathematical expressions to which the same operations apply as the operations applied to the indeterminate.

The concept of an indeterminate is relatively recent, and was initially introduced for distinguishing a polynomial from its associated polynomial function.^{[citation needed]} Indeterminates resemble free variables. The main difference is that a free variable is intended to represent an unspecified element of some domain, often the real numbers, while indeterminates do not represent anything.^{[citation needed]} Many authors do not distinguish indeterminates from other sorts of variables.

Some authors of abstract algebra textbooks define an indeterminate over a ring R as an element of a larger ring that is transcendental over R.