Answer to Question #179097 in Abstract Algebra for Anand

Question

Check whether or not Q[x] / < 8x³ + 6x² − 9x + 24 > is a field

Answer 1

With p = 3, a₀ ≡ a₁ ≡ a₂ ≡ 0 (mod p); thus 24 ≡ −9 ≡ 6 ≡ 0(mod 3).

a_n = a₃ !≡ 0 (mod 3) since a₃ = 8 ≡ 2 (mod 3)

a₀ !≡ 0 (mod p²) since a₀ = 24 ≡ 6 (mod 9).

Therefore, by Eisenstein Criterion, 8x³ + 6x² − 9x + 24 is irreducible over Q. Thus the quotient ring is a field