Does Eisenstein's criterion apply when some coefficients are $0$?

Question

I've just been reading the Wikipedia page on Eisenstein's criterion; in summary it says that the polynomial with integer coefficients $$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$ is irreducible over the rationals if there exists a prime number $p$ such that

• $p$ divides each $a_i$ for $0\leqslant i<n$
• $p$ does not divide $a_n$ and
• $p^2$ does not divide $a_0$

I've just been wondering: does Eisenstein's criterion also apply even when some of the coefficients are $0$? Do we consider $p$ to always divide $0$?

Thank you for your help.

Answer 1

A number $a$ divides a number $b$ if there exists some value $x$ such that $a\cdot x = b$.

By definition, all numbers divide the number $0$, since the value $x=0$ satisfies the condition for all of them.

Answer 2

Both in general, and in the context of Eisenstein in particular, we always have $n$ divides $0$ for any integer $n$.

The definition of $a\mid b$ is

There is an integer $x$ such that $b=xa$.

For $b=0$, it always works to set $x=0$, no matter what $a$ is.