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!