Indeterminate (variable)

In mathematics, particularly modern algebra, an indeterminate is a variable that is used formally, without reference to any value, although, in some contexts, an indeterminate is considered to be its own value.^{[citation needed]} In other words, this is is just a symbol used in a formal way, which has no other value than itself.^[how?]

Indeterminates occur in polynomials, formal power series, and, sometimes, in expressions that are viewed as independent mathematical objects.^[1]

The concept of an indeterminate is relatively recent, and was initially introduced for distinguishing a polynomial from its associated polynomial function. 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. Many authors do not distinguish indeterminates from other sorts of variables.

Some authors use the term to mean generaly "variable".^[2]^[3]^[4]^[5]^[6] One attempt to formally define the term in this way consideres an indeterminate to be the generator of an infinite cyclic group.^[7]^[a] Other attempts define an indeterminate over a ring $R$ as an element of a larger ring that is transcendental over $R$ .^[8]^[9]^[10] That is, an element $X$ is an indeterminate if and only if, $a_{0}+a_{1}X+a_{2}X^{2}+\dotsm a_{n}X^{n}=0$ implies that each $a_{i}=0,{\text{ }}0\leq i\leq n$ , where each coefficient is an element in $R$ . This definition implies that every real, transcendental number, like $π$ and $e$ , are considered as indeterminates over the rationals.

Other authors use it as a kind of syntactic entity, similar to punctuation, in that it does not represent anything or have any algebraic properties.^[11]^[12]^[13]^[14] Many authors who attempt to formalize the term indeterminate by this definition do so by, for example, defining the expression $3+5X+2X^{2}$ to mean the ordered tuple $(3,5,2)$ .

Polynomials

A polynomial in an indeterminate $X$ is an expression of the form $a_{0}+a_{1}X+a_{2}X^{2}+\ldots +a_{n}X^{n}$ , where the $a_{i}$ are called the coefficients of the polynomial. Two such polynomials are equal only if the corresponding coefficients are equal.^[15] In contrast, two polynomial functions in a variable $x$ may be equal or not at a particular value of $x$ .

For example, the functions

f(x)=2+3x,\quad g(x)=5+2x

are equal when $x=3$ and not equal otherwise. But the two polynomials

2+3X,\quad 5+2X

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

2+3X=a+bX

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

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

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

so the polynomial function $x-x^{2}$ is identically equal to 0 for $x$ having any value in the modulo-2 system. However, the polynomial $X-X^{2}$ 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 $X$ is an expression of the form $a_{0}+a_{1}X+a_{2}X^{2}+\ldots$ , where no value is assigned to the symbol $X$ .^[16] 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 $x$ , such as $1+x+2x^{2}+6x^{3}+\ldots +n!x^{n}+\ldots \,$ , are allowed.

As generators

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

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

Notes

^ McCoy (1960, pp. 189, 190)
^ Vivaldi, Franco (2014). Mathematical writing. New York : Springer. ISBN 978-1-4471-6526-2.{{cite book}}: CS1 maint: publisher location (link)
^ Murty, Maruti Ram (2014). Transcendental numbers. New York : Springer. ISBN 978-1-4939-0831-8.{{cite book}}: CS1 maint: publisher location (link)
^ Hackley, Charles W. (Charles William) (1846). A Treatise on Algebra, Containing the Latest Improvements. University of California Libraries. New York, Harper & Brothers.
^ Becker, Thomas (1993). Gröbner bases : a computational approach to commutative algebra. New York : Springer-Verlag. ISBN 978-0-387-97971-7.
^ Joseph Miller Thomas (1974). A Primer On Roots. William Byrd Press. ASIN B0006W3EBY.
^ Lang, Serge (1993). Algebra (PDF). New York: Springer-Verlag. p. 97. ISBN 0-387-95385-X.
^ Lewis, Donald J. (1965). Introduction to Algebra. New York: Harper & Row. p. 160. LCCN 65-15743.
^ Landin, Joseph (1989). An Introduction to Algebraic Structures. New York: Dover Publications. p. 204. ISBN 0-486-65940-2.
^ Marcus, Marvin (1978). Introduction to Modern Algebra. New York: Marcel Dekker. p. 140–141. ISBN 0-8247-6479-X.
^ Sawyer, W. W. (2018). A Concrete Approach to Abstract Algebra. San Francisco: Dover Publications. ISBN 978-0486824611.
^ Sah, Chih-Han (1967). Abstract algebra. Academic Press Textbooks in Mathematics. ISBN 978-0126147506.{{cite book}}: CS1 maint: date and year (link)
^ Pinter, Charles C. (2010). A book of abstract algebra. Mineola, N.Y. : Dover Publications. ISBN 978-0-486-47417-5.
^ Herstein, I. N. (1975). Topics in algebra. New York : Wiley. ISBN 978-0-471-01090-6.{{cite book}}: CS1 maint: publisher location (link)
^ Herstein 1975, Section 3.9 harvnb error: multiple targets (2×): CITEREFHerstein1975 (help).
^ Weisstein, Eric W. "Formal Power Series". mathworld.wolfram.com. Retrieved 2019-12-02.

^ Lang does not use the term indeterminate in his book, however, he does state that the symbol X is a "variable" in quotes, to emphasize that X is not a variable in the usual sense.

References

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

[1] McCoy (1960, pp. 189, 190)

[2] Vivaldi, Franco (2014). Mathematical writing. New York : Springer. ISBN 978-1-4471-6526-2.{{cite book}}: CS1 maint: publisher location (link)

[3] Murty, Maruti Ram (2014). Transcendental numbers. New York : Springer. ISBN 978-1-4939-0831-8.{{cite book}}: CS1 maint: publisher location (link)

[4] Hackley, Charles W. (Charles William) (1846). A Treatise on Algebra, Containing the Latest Improvements. University of California Libraries. New York, Harper & Brothers.

[5] Becker, Thomas (1993). Gröbner bases : a computational approach to commutative algebra. New York : Springer-Verlag. ISBN 978-0-387-97971-7.

[6] Joseph Miller Thomas (1974). A Primer On Roots. William Byrd Press. ASIN B0006W3EBY.

[7] Lang, Serge (1993). Algebra (PDF). New York: Springer-Verlag. p. 97. ISBN 0-387-95385-X.

[9] Lewis, Donald J. (1965). Introduction to Algebra. New York: Harper & Row. p. 160. LCCN 65-15743.

[10] Landin, Joseph (1989). An Introduction to Algebraic Structures. New York: Dover Publications. p. 204. ISBN 0-486-65940-2.

[11] Marcus, Marvin (1978). Introduction to Modern Algebra. New York: Marcel Dekker. p. 140–141. ISBN 0-8247-6479-X.

[12] Sawyer, W. W. (2018). A Concrete Approach to Abstract Algebra. San Francisco: Dover Publications. ISBN 978-0486824611.

[13] Sah, Chih-Han (1967). Abstract algebra. Academic Press Textbooks in Mathematics. ISBN 978-0126147506.{{cite book}}: CS1 maint: date and year (link)

[14] Pinter, Charles C. (2010). A book of abstract algebra. Mineola, N.Y. : Dover Publications. ISBN 978-0-486-47417-5.

[15] Herstein, I. N. (1975). Topics in algebra. New York : Wiley. ISBN 978-0-471-01090-6.{{cite book}}: CS1 maint: publisher location (link)

[16] Herstein 1975, Section 3.9 harvnb error: multiple targets (2×): CITEREFHerstein1975 (help).

[17] Weisstein, Eric W. "Formal Power Series". mathworld.wolfram.com. Retrieved 2019-12-02.

[8] Lang does not use the term indeterminate in his book, however, he does state that the symbol X is a "variable" in quotes, to emphasize that X is not a variable in the usual sense.

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Polynomials

Formal power series

As generators

See also

Notes

References