Degree of the splitting field over finite field

Question

Let the K be a splitting field of $f(x) =x^3 + 5x+ 5$ over $Z_3 $

What is the $[K;Z_3]$ ?

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The solution in my book)

Let the $\alpha$ be a solution of the $f(x$)

Since $K = Z_3 (\alpha$) Therefore, $[K; Z_3] = 3$

But My question

It splitting field surely have the other solution $\beta$ and $\gamma$

Hence $K = Z_3(\alpha, \beta, \gamma)$ Not the $Z_3(\alpha)$

So does it have to be $[K;Z_3] \geq 3$ ?

(I.e. Can't sure that $[K;Z_3] = 3$)

Answer 1

Well, if you consider an irreducible polynomial $f(x)$ of degree $n\geq 1$ over $GF(p)$, then the finite field $GF(p^n) = GF(p)[x]/\langle f(x)\rangle$ is an extension of $GF(p)$ of degree $n$ and if $\alpha\in GF(p^n)$ is a zero of $f(x)$, then $\alpha,\alpha^p,\ldots,\alpha^{p^{n-1}}$ are the zeros of $f(x)$.

Answer 2

You'll find that if $\alpha$ is a root then the other roots are $\alpha^3$ and $\alpha^9$.

Answer 3

Any finite extension of a finite field is a normal extension. Since $f$ is an irreducible polynomial in $Z_3$ and $Z_3(\alpha)/Z_3$ is a normal extension $f$ must split in $Z_3(\alpha)$.