Opposite Eisenstein's criteria

[lola1990]

Homework Statement

Homework Equations

The Attempt at a Solution

 

[micromass]

Note, if you want to type in TeX, then you need to use the / slash. So it would have to be [ itex ] ... [ /itex ]

Homework Statement

Let $f(x)=a_{n}x^{n}+...+a_{1}x+a_{0}$. Let p be a prime and suppose $p~|~ a_{i}$ for i in n,n-1...1 but p does not divide $a_{0}$. Show that if $p^{2}$ does not divide $a_{n}$, then f(x) is irreducible.

Homework Equations

The Attempt at a Solution

Let f(x)=h(x)g(x) with h(x),g(x) in Z[x], and reduce mod p so that $a_{0}=h(x)g(x)$. We have that if the leading coefficient of g(x) is $g_{r}$ and the leading term of h(x) is $h_{s}$ with r+s=n, p divides either coefficient but not both (because then the product would be divisible by $p^{2}$). Also, p does not divide the constant term of either polynomial. WLOG, suppose p divides $g_{r}$ but not $h_{s}$. Now, I want to find a coefficient of f(x) so that I can force $h_{s}$ to be divisible by p, but I'm not sure how... help!