Indeterminate (variable)

In mathematics, and particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, that does not stand for anything else except itself and is often used as a placeholder in objects such as polynomials and formal power series.[1][2][3] In particular:

  • It does not designate a constant or a parameter of the problem.
  • It is not an unknown that could be solved for.
  • It is not a variable designating a function argument, or a variable being summed or integrated over.
  • It is not any type of bound variable.
  • It is just a symbol used in an entirely formal way.[4]

Polynomials

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.[5] In contrast, two polynomial functions in a variable may be equal or not depending on the 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 series

A formal power series in an indeterminate is an expression of the form , where no value is assigned to the symbol .[6] 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 generators

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 also

Notes

  1. "The Definitive Glossary of Higher Mathematical Jargon — Indeterminate". Math Vault. 2019-08-01. Retrieved 2019-12-02.
  2. Weisstein, Eric W. "Indeterminate". mathworld.wolfram.com. Retrieved 2019-12-02.
  3. "Definition:Polynomial Ring/Indeterminate - ProofWiki". proofwiki.org. Retrieved 2019-12-02.
  4. McCoy (1973, pp. 189,190)
  5. Herstein 1975, Section 3.9.
  6. Weisstein, Eric W. "Formal Power Series". mathworld.wolfram.com. Retrieved 2019-12-02.

References

  • Herstein, I. N. (1975). Topics in Algebra. Wiley.
  • McCoy, Neal H. (1973), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68015225

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