Let $f(x) = x^3 - 3x^2 + 15x - 6 \in Q[x]$ and $I = (f) \in Q[x]$.
Prove that the quotient ring $Q[x]/I$ is a field.
Since $Q[x]$ is a PID it suffices to show that $f$ is irreducible, right? how one can prove that?
2026-03-29 20:55:02.1774817702
Proving a quotient ring is a field.
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By the rational root theorem, $f$ has no rational roots. Since the degree is $3$, it follows that $f$ is irreducible. Eisenstein criterion with $p=3$ is another way to show this.