Division of two polynomial expressions

Question

Is $$1/(X^n - 1), n \in N$$ a polynomial?

Intuitively I would say yes, because 1 is a polynomial($ X^0$) and so is $X^n - 1$. But Sage (The CAS) appears to disagree, when I type the expression in and call the function is_polynomial() I get False. Can somebody explain why this expression isn't a polynomial?

Answer 1

Polynomials in $x$ are defined as a finite sum of terms of the form $ax^n$. $a$ is restricted to some set like the integers, rationals, reals, integers $\bmod $ something, complex numbers. $n$ is restricted to the nonnegative integers. Rational functions of polynomials do not count, so Sage is correct.

Answer 2

It is a rational fraction, i.e. the quotient of two polynomials in the field of fractions of the polynomial ring $K[X]$ ($K$ is the field of coefficients, which you did not specify). This field of fractions is denoted $K(X).$