Indeterminate (variable)

In mathematics, particularly abstract algebra, an indeterminate is a variable symbol used to describe a class of expressions over a given ring, usually to define a new ring over these expressions. For example, a polynomial over a given ring $R$, may be defined as an expression of the form:$$a_{0}+a_{1}X+a_{2}X^{2}+...a_{n}X^{n}$$Where, the coefficients ${\textstyle a_{i},\;0\leq i\leq n,}$ are elements of $R$, and $X$ is an indeterminate (which is not considered an element of $R$). Then, one may define the ring of polynomials over $R$ in the indeterminate $X$, usually denoted $R[X]$. Indeterminates are often used in defining rings over polynomials, algebraic fractions, formal power series, and other algebraic expressions.

There are generally two camps that attempt to formalize this term. The first takes an indeterminates to be a kind of purely syntactic entity, where the indeterminate $X^{k}$ (for some ⁠$k\leq n$⁠) simply tells you that a coefficient occures in the kth position. In this sense, many authors define the polynomial ${\textstyle a_{0}+a_{1}X+a_{2}X^{2}+...a_{n}X^{n}}$ to be the infinite sequence $(a_{0},\,a_{1},\,a_{2},\,...\,a_{n},\,...)$ where all $a_{i}=0,\;i>n$. By this definition, it is clear that $X$ is not, and does not denote, any kind of object.

In the second camp, one takes an indeterminate to denote an actual object with algebraic properties, specifically, given a ring $R$, an element $X$ (not in $R$) is an indeterminate over $R$ if and only if it is transcendental over $R$; that is, ${\textstyle \sum _{i=0}^{n}a_{i}X^{i}=0}$ if and only if $a_{i}=0,\;0\leq i\leq n$; and sometimes is required to commute with all elements of $R$. By this definition, the real number π is an indeterminate over the rationals, but ${\sqrt {3}}$ is not, since $3-1({\sqrt {3}})^{2}=0$. The diffrence between an indeterminate and a transcendental element is in intension: the term "indeterminate" is reserved for when one is thinking of evaluating the indeterminate symbol. In the context of formal power series, the term "indeterminate" is generalized to be "transcendental" over power series. In this sense, π is not an indeterminate (for instance, using the Maclaurin series of sine), but the construction in the following paragraph is.

Some authors offer the specific construction if an infinite coordinate vector (or sequence) as: $X^{0}=(1,0,0,...)$, $X^{1}=(0,1,0,...)$, and in general, $X^{n}$ is the sequence with the nth position having 1, and all others 0.

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 aplied to the indeterminate.

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.

Polynomials

Main article: Polynomial

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. 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

Main article: 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 $X$. 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

Main article: Generator (mathematics)

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).

See also

• Indeterminate equation
• Indeterminate form
• Indeterminate system

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