We are familiar with the problem that $$f(x)=(x-a_1)(x-a_2)\cdots(x-a_n)\pm1$$
and $$f(x)=(x-a_1)^2(x-a_2)^2\cdots(x-a_n)^2+1$$
is irreducible with $a_i\in \mathbb{Z}$ and for the first problem, when $\pm \mapsto +$,it should be $n\neq 2,4$.
I know that R.Brauer has proven that $$f(x)=A(x-a_1)^4(x-a_2)^4\cdots(x-a_n)^4+1$$
is irreducible whenever $A>0,A\neq 4k^4,k\in\mathbb{Z}$. I tried some thoughts and then no ideas sprang. So I wonder how to prove it.