Hi, i search for irreducible polynomials over Z which have variable coefficients you can "choose". Since I found nearly nothing in books or the internet i hope you can help me. Here 3 examples: Let g a polynomial over Z with degree smaller then n/2 ,then: $g* (\prod_{i=1}^n (x-a_i)) -1 $ is irreducible if the a_i are all distinct. Here you can choose n the coefficients of g and the a_n so its a nice example. Another one,I found, is from Furtwängler : $x^4 (\prod_{i=1}^{n-4} (x-b_i)) -(-1)^n *(2x+4) $ where the b_i are strictly increasing.Can you generalize this example ?I think it should work also for some integers other than 2 and 4. Here is another nice example : Polynomial with the primes as coefficients irreducible?

I try to find examples where its easy to control zeros modulo p of the some irreducible polynomials and its derivation for another problem.