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

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?

## marked as duplicate by lulu, Community♦Nov 4 '18 at 0:26

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 $$(x+2)(x+3)$$.