Indeterminate (variable)

In mathematics, particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else except itself. It may be used as a placeholder in objects such as polynomials and formal power series.[1][2] In particular:

When used as placeholders, a common operation is to substitute mathematical expressions (of an appropriate type) for the indeterminates.

By a common abuse of language, mathematical texts may not clearly distinguish indeterminates from ordinary variables.


A polynomial in an indeterminate   is an expression of the form  , where the   are called the coefficients of the polynomial. Two such polynomials are equal only if the corresponding coefficients are equal.[4] In contrast, two polynomial functions in a variable   may be equal or not at a particular value of  .

For example, the functions


are equal when   and not equal otherwise. But the two polynomials


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


does not hold unless   and  . This is because   is not, and does not designate, a number.

The distinction is subtle, since a polynomial in   can be changed to a function in   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:


so the polynomial function   is identically equal to 0 for   having any value in the modulo-2 system. However, the polynomial   is not the zero polynomial, since the coefficients, 0, 1 and −1, respectively, are not all zero.

Formal power seriesEdit

A formal power series in an indeterminate   is an expression of the form  , where no value is assigned to the symbol  .[5] 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  , such as  , are allowed.

As generatorsEdit

Indeterminates are useful in abstract algebra for generating mathematical structures. For example, given a field  , the set of polynomials with coefficients in   is the polynomial ring with polynomial addition and multiplication as operations. In particular, if two indeterminates   and   are used, then the polynomial ring   also uses these operations, and convention holds that  .

Indeterminates may also be used to generate a free algebra over a commutative ring  . For instance, with two indeterminates   and  , the free algebra   includes sums of strings in   and  , with coefficients in  , and with the understanding that   and   are not necessarily identical (since free algebra is by definition non-commutative).

See alsoEdit


  1. ^ Weisstein, Eric W. "Indeterminate". Retrieved 2019-12-02.
  2. ^ "Definition:Polynomial Ring/Indeterminate - ProofWiki". Retrieved 2019-12-02.
  3. ^ McCoy (1973, pp. 189, 190)
  4. ^ Herstein 1975, Section 3.9.
  5. ^ Weisstein, Eric W. "Formal Power Series". Retrieved 2019-12-02.


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