Ring Theory Final, Problem 1

1) Let \( x \) denote an indeterminate, and consider the ring of polynomials \( \mathbb{Z}[x] \) with integer coefficients. Determine if the following polynomial is irreducible in \( \mathbb{Z}[x]: \) \[ f(x) = x^6 + 30 x^5 - 15 x^3 + 6x -120. \]

Proof. \( f(x) \) is a monic polynomial and the prime \( 3 \) divides every other coefficient, but \( 3^2 = 9 \) does not divide the constant term (\( -120 \)). Therefore, by Eisenstein's Criterion, \( f(x) \) is irreducible over \( \mathbb{Z}[x]. \) QED.

Note: This is part of Ex. §9.4,#2 in Dummit & Foote.