# Prove that $(X^2 + 1)^n + p$ is irreducible over $\mathbb{Q}[X]$

Let $$p$$ an odd prime number, congruent to $$3$$ mod $$4$$. Prove that the polynomial $$f(x) = (X^2 + 1)^n + p$$ is irreducible over the ring $$\mathbb{Q}[X]$$, regardless of the value of $$n$$ (natural number).

Doing some research, I have found a similar thread in AoPS community, that asks about a similar question (link: https://artofproblemsolving.com/community/c6h1455017p8368425) about the same polynomial being irreducible over $$\mathbb{Z}[X]$$. My specific questions are:

1. If I know that a polynomial is irreducible in $$\mathbb{Z}[X]$$, can we deduce something about being irreducible in $$\mathbb{Q}[X]$$. What conditions should we impose such that this transition may be made. What about the converse one?

2. What is the ring $$\mathbb{F}_p[X]$$? What is the relationship between this ring and $$\mathbb{Z}[X]$$ and $$\mathbb{Q}[X]$$?

3. If none of the concepts stated have any relevance, how can we approach this problem? In particular, how to apply irreducibility criteria to this type of problem?

• There are not too many ways in which one can detect irreducibility of polynomials by hand. If I see a (and only one) prime $p$ then I'm trying to reduce mod $p$ for sure, and see if $f$ is irreducible modulo $p$. If not, then I'm trying to contradict the definition of reducibility by finding structure in the factors of $f$. In this case, reducing modulo $p$ provides structure to possible factors of $f$ which aids in proving irreducibility. Jan 27 at 17:59

As you know (from the link) that $$f$$ is irreducible in $$\mathbb Z[x]$$, the irreducibility of $$f$$ in $$\mathbb Q[x]$$ is a direct consequence of Gauss's lemma for polynomials and the fact that the leading coefficient of $$f$$ is equal to $$1$$.
$$\mathbb F_p[x]$$ is the ring of polynomials over the finite field $$\mathbb F_p$$.