Let $K$ be a field with characteristic $p$ and the polynomial $P(x)= x^{2023}+x+1 \in K[x]$. Let $E$ be the splitting field of $P(x)$ over $K$. Prove that $E/K$ is a cyclic extension.
My progress so far: This is a problem in my final term two days ago. A given hint is to compare $E$ with the splitting field of $P(x)$ over $\mathbb{F}_p$. If $K$ is also a finite field, then the problem is reduced to prove that a finite extension of finite field is cyclic and I could do that. But I wonder how to approach when $K$ is an infinite field, for e.g: $K=\mathbb{F}_p(t)$.
I guess that there is a specific field homomorphism $\phi$ - just like the Frobenius in the case when $K = \mathbb{F}_p$ and show that $\text{Gal}(E/K)$ is generated but such $\phi$, but so far I can't think of any maps like that. Any hints are really appreciated.
Hint: Let $\alpha_1,...,\alpha_n$ be the roots of the polynomial in some fixed algebraic closure of $K$. Then $E=K(\alpha_1,...,\alpha_n)$. Define:
$$\varphi:\text{Gal}(K(\alpha_1,...,\alpha_n)/K)\to\text{Gal}(\mathbb{F_p}(\alpha_1,...,\alpha_n)/\mathbb{F_p})$$
by restriction $\sigma\to\sigma|_{\mathbb{F_p}(\alpha_1,...,\alpha_n)}$. Verify that this is a well defined (i.e the restriction is indeed an automorphism of $\mathbb{F_p}(\alpha_1,...,\alpha_n)$), injective group homomorphism.