Questions concerning $\mathbb Z_3[x]/(x^3+2x-1)$

Question

Is the automorphism group of $\mathbb Z_3[x]/(x^3+2x-1)$ cyclic ? Is $\mathbb Z_3[x]/(x^3+2x-1)$ separable ?

Answer 1

Notice that $$ \frac{\mathbb{Z}_3[x]}{(x^3+2x-1)}\cong \mathbb{F}_{27} $$ So that both statements are true:

• Its automorphisms form a cyclic group, its elements are generated by the Frobenius automorphism.
• To see that it's seperable (I assume you mean that it's separable over $\mathbb{Z}_3$) we have to prove that every element of $\mathbb{F}_{27}$ has a minimal polynomial in $\mathbb{Z}_3$ which has no multiple roots. And this is quite obviously true, because the degree of the minimal polynomial is at most $3$, so it's derivative will only vanish if the polynomial is of the form $X^3+a = (X+a)^3$, but this polynomial is not irreducible. Does this makes sense? I assumed you already had a little experience with seperability.