# In $\mathbb Z[x_1,x_2,\ldots,x_n]$, show $x_1 x_2\cdots x_n$ has $2^{n+1}-2$ non-constant polynomials dividing it

This particular question is a part of an assignment which could not be discussed due to pandemic .

Question (a) Consider the ring R of polynomials in n variables with integer coefficients . Prove that the polynomial $$f(x_1 , x_2 ,\ldots, x_n) = x_1 x_2\cdots x_n$$ has $$2^{n+1}-2$$ non-constant polynomials in R dividing it .

Attempt: (a) any polynomial in R dividing f would be of form $${x_{1}}^{i} ...{x_{n}}^{i}$$ i=0 or 1. So , there will be $$2^{n}-1$$ polynomials . But the answer is 2 times my answer.

Thanks!!

• Hint for (a): does $-x$ divide $xy$? Sep 2, 2020 at 17:30
• And also after you split, you should be easily able to make titles that aren't useless. Sep 2, 2020 at 18:04

Note that if $$x_i ...x_k$$ divides f then - ($$x_i ...x_k$$) also divides f . So, correct answer is $$2^{n+1}-2$$.
In a) you forgot the coefficient: $$-x_1$$ divides $$x_1x_2$$. This gives $$2^{n+1}-2$$ divisors because the coefficient can be $$1$$ or $$-1$$.