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

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 $$0\leqslant i
• $$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 $$0$$?

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.

• Thank you for the clarification! Commented Mar 24, 2021 at 14:35

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.

• Thank you for the clarification! Commented Mar 24, 2021 at 14:35