Indeterminate (variable)

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In mathematics, an indeterminate is a variable that is used formally, without reference to any value. In other words, this is just a symbol used in a formal way.^{[1][2][better source needed]} Indeterminates occur in polynomials, formal power series, and, more generally, in expressions that are viewed as independent mathematical objects.

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 aplied 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 a 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.^[3][4][5] This uncommon definition implies that every transcendental number and every nonconstant polynomial must be considered as indeterminates.

A polynomial in an indeterminate ${\displaystyle X}$ is an expression of the form ${\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\ldots +a_{n}X^{n}}$, where the ${\displaystyle a_{i}}$ are called the coefficients of the polynomial. Two such polynomials are equal only if the corresponding coefficients are equal.^[6] In contrast, two polynomial functions in a variable ${\displaystyle x}$ may be equal or not at a particular value of ${\displaystyle x}$.

For example, the functions

${\displaystyle f(x)=2+3x,\quad g(x)=5+2x}$

are equal when ${\displaystyle x=3}$ and not equal otherwise. But the two polynomials

${\displaystyle 2+3X,\quad 5+2X}$

are unequal, since 2 does not equal 5, and 3 does not equal 2. In fact,

${\displaystyle 2+3X=a+bX}$

does not hold unless ${\displaystyle a=2}$ and ${\displaystyle b=3}$. This is because ${\displaystyle X}$ is not, and does not designate, a number.

The distinction is subtle, since a polynomial in ${\displaystyle X}$ can be changed to a function in ${\displaystyle x}$ by substitution. But the distinction is important because information may be lost when this substitution is made. For example, when working in modulo 2, we have that:

${\displaystyle 0-0^{2}=0,\quad 1-1^{2}=0,}$

so the polynomial function ${\displaystyle x-x^{2}}$ is identically equal to 0 for ${\displaystyle x}$ having any value in the modulo-2 system. However, the polynomial ${\displaystyle X-X^{2}}$ is not the zero polynomial, since the coefficients, 0, 1 and −1, respectively, are not all zero.

A formal power series in an indeterminate ${\displaystyle X}$ is an expression of the form ${\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\ldots }$, where no value is assigned to the symbol ${\displaystyle X}$.^[7] This is similar to the definition of a polynomial, except that an infinite number of the coefficients may be nonzero. Unlike the power series encountered in calculus, questions of convergence are irrelevant (since there is no function at play). So power series that would diverge for values of ${\displaystyle x}$, such as ${\displaystyle 1+x+2x^{2}+6x^{3}+\ldots +n!x^{n}+\ldots \,}$, are allowed.

Indeterminates are useful in abstract algebra for generating mathematical structures. For example, given a field ${\displaystyle K}$, the set of polynomials with coefficients in ${\displaystyle K}$ is the polynomial ring with polynomial addition and multiplication as operations. In particular, if two indeterminates ${\displaystyle X}$ and ${\displaystyle Y}$ are used, then the polynomial ring ${\displaystyle K[X,Y]}$ also uses these operations, and convention holds that ${\displaystyle XY=YX}$.

Indeterminates may also be used to generate a free algebra over a commutative ring ${\displaystyle A}$. For instance, with two indeterminates ${\displaystyle X}$ and ${\displaystyle Y}$, the free algebra ${\displaystyle A\langle X,Y\rangle }$ includes sums of strings in ${\displaystyle X}$ and ${\displaystyle Y}$, with coefficients in ${\displaystyle A}$, and with the understanding that ${\displaystyle XY}$ and ${\displaystyle YX}$ are not necessarily identical (since free algebra is by definition non-commutative).

• Indeterminate equation
• Indeterminate form
• Indeterminate system

• Herstein, I. N. (1975). Topics in Algebra. Wiley. ISBN 047102371X.
• McCoy, Neal H. (1960), Introduction To Modern Algebra, Boston: Allyn and Bacon, LCCN 68015225