Use Eisenstein´s criterion to show that $P(x)=x^n+5x^{n-1}+3$ is irreducible.

Question

Since Eisenstein´s criterion ist not directly applicable we look at the polynomial P (mod 5), which then reduces to $P(x) = x^n + 3$ (mod 5) ... is this correct? Now I can apply the Eisenstein Criterion with prime number 3: The leading coefficient of $x^n$ is 1 and not divisible by 3. The other coefficients, 0 and 3 are divisible by the prime 3. Further, the constant, 3, ist not divisible by the square of the chosen prime, $3^2 = 9$. Therefore, $P$ is an irreducible polynomial. Is this correct? Have I understood the E.C. correctly?

Answer 1

No, this is not correct. Eisenstein's criterion is for polynomials in $\mathbb{Z}[x]$, not in $\mathbb{Z}_p[x]$, for some prime $p$. For instance, by your argument, $x^2+5x+6$ would be irreducible. But it is reducible, since it is equal to $(x+2)(x+3)$.