Indeterminate (variable) explained

In mathematics, an indeterminate or formal variable is a variable (a symbol, usually a letter) that is used purely formally in a mathematical expression, but does not stand for any value.^[1]

In analysis, a mathematical expression such as is usually taken to represent a quantity whose value is a function of its variable, and the variable itself is taken to represent an unknown or changing quantity. Two such functional expressions are considered equal whenever their value is equal for every possible value of within the domain of the functions. In algebra, however, expressions of this kind are typically taken to represent objects in themselves, elements of some algebraic structure – here a polynomial, element of a polynomial ring. A polynomial can be formally defined as the sequence of its coefficients, in this case, and the expression or more explicitly is just a convenient alternative notation, with powers of the indeterminate used to indicate the order of the coefficients. Two such formal polynomials are considered equal whenever their coefficients are the same. Sometimes these two concepts of equality disagree.

Some authors reserve the word variable to mean an unknown or changing quantity, and strictly distinguish the concepts of variable and indeterminate. Other authors indiscriminately use the name variable for both.

Indeterminates occur in polynomials, rational fractions (ratios of polynomials), formal power series, and, more generally, in expressions that are viewed as independent 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 applied to the indeterminate.

Some authors of abstract algebra textbooks define an indeterminate over a ring as an element of a larger ring that is transcendental over .^{[2] [3] [4]} This uncommon definition implies that every transcendental number and every nonconstant polynomial must be considered as indeterminates.

Polynomials

See main article: Polynomial. A polynomial in an indeterminate

is an expression of the form , where the

a_i

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

x

may be equal or not at a particular value of

x

.

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

See main article: Formal power series. A formal power series in an indeterminate

is an expression of the form

a₀+a_1X+

	2
a
	2X

+\ldots

, 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

See main article: Generator (mathematics). 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

• Indeterminate equation
• Indeterminate form
• Indeterminate system

References

• Book: 047102371X. Topics in Algebra. Herstein. I. N.. 1975. Wiley .

Notes and References

1. Book: Joseph Miller Thomas . A Primer On Roots . 1974 . William Byrd Press . B0006W3EBY.
2. Book: Lewis, Donald J. . Introduction to Algebra . . 1965 . New York . 65-15743. 160.
3. Book: Landin, Joseph . An Introduction to Algebraic Structures . . 1989 . 0-486-65940-2 . New York. 204.
4. Book: Marcus, Marvin . Introduction to Modern Algebra . . 1978 . 0-8247-6479-X . New York. 140–141.
5. .
6. Web site: Formal Power Series. Weisstein. Eric W.. mathworld.wolfram.com. en. 2019-12-02.