Irreducible over $\mathbb{Q}$ (ring $\mathbb{Z}$)

Question

How to prove the irreduciblity over $\mathbb{Q}$ (ring $\mathbb{Z}$) of polynomials $$f = (x-{a_1})\cdots(x-{a_n})-1$$ and $$g = (x-{a_1})^2\cdots(x-{a_n})^2+1,$$ {${a_i}$} - pairwise distinct integers.

Answer 1

Suppose

$$f(x)=h(x)k(x)\;,\;\;h(x),k(x)\in\Bbb Z[x].\text{ But since }\;f(a_i)=-1=h(a_i)k(a_i)\;\;\text{we get}$$

$$ h(a_i)=-k(a_i)=\pm1\;,$$

for all $\;a_1,...,a_n\;$. But then $\;h(x)+k(x)\;$ is a polynomial of degree less than $\;n\;$ (otherwise the above decomposition of $\;f(x)\;$ is not a true one) which vanishes in $\;n\;$ different points , so

$$h(x)+k(x)\equiv0\implies h(x)=-k(x)\implies f(x)=-h(x)^2$$

but this last equality is impossible by comparing the coefficients of $\;x^n\;$ in both sides.

Now you try the other one.