Indeterminate (variable)
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In mathematics, particularly in modern algebra, an indeterminate is a variable symbol that is used to classify expressions of a certain type over some Ring or other algebraic structure.[1][2] The difference between a variable and an indeterminate is that the latter cannot be assigned a value as it does not represent any particular class of objects. In this way, an indeterminate can be defined as a variable with no inherent domain.
For instance, a polynomial in the ring of polynomials may be defined as an expression of the form:
For some natural number , where each coefficient is taken as a parameter, and is an indeterminate, together with standard properties of addition and multiplication. Specifically, distributivity and the ability to combine like terms.
This declaration is useful as it generalizes the algebraic properties of whatever expression one is working with. Specifically, any ring with variables over some specified domain can be said to be isomorphic to that ring over indeterminates. However, it is a common abuse of language that mathematical texts may not clearly distinguish indeterminates from ordinary variables.
It 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. In particular:
- It does not designate a constant or a parameter of the problem.
- It is not any type of bound variable, for instance designating a function argument, or being summed or integrated over
- It is just a symbol used in an entirely formal way.[3]
When used as placeholders, a common operation is to substitute mathematical expressions (of an appropriate type) for the indeterminates.
Polynomials
[edit]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 series
[edit]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 generators
[edit]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
[edit]Notes
[edit]- ^ Weisstein, Eric W. "Indeterminate". mathworld.wolfram.com. Retrieved 2019-12-02.
- ^ "Definition:Polynomial Ring/Indeterminate - ProofWiki". proofwiki.org. Retrieved 2019-12-02.
- ^ McCoy (1973, pp. 189, 190)
- ^ Herstein 1975, Section 3.9.
- ^ Weisstein, Eric W. "Formal Power Series". mathworld.wolfram.com. Retrieved 2019-12-02.
References
[edit]- Herstein, I. N. (1975). Topics in Algebra. Wiley. ISBN 047102371X.
- McCoy, Neal H. (1973), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68015225
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