Find all values of such that the following polynomial is irreducible: .
Suppose is reducible. Then it factors as a product of linear polynomials. By Gauss’ lemma, we can assume without loss of generality that these factors are in . Since the leading coefficient of is prime, we can say (again without loss of generality) that . Comparing coefficients, we have and . the first equation has 12 solutions in the integers: , , , , , and . Reducing the second equation mod 2, we see that mod 2. This reduces our set of candidate solutions to four. If , then and indeed . If , then and indeed . If , then and indeed . If , then , and indeed .
For all other , is irreducible.