Answer to Question #179097 in Abstract Algebra for Anand

Question #179097

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


1
Expert's answer
2021-05-03T05:54:47-0400

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

an = 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


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