Exercise 8.1.8 (Galois and Fix not mutually inverse)

Solution: We have that $M=S{F}_K(t^p-u)$ and $K={𝔽}_p(u)$, hence as in Example 7.2.19 (ii) we have that $[M:K]=\mathit{de}{g}_K(\alpha )=p$, where $p$ is prime. This means that $\left|𝔉\right|=p$, and hence by the tower law we know that there are no trivial intermediate fields, hence $𝔉=\{M,K\}$. We have that $|\mathit{Gal}(M:K)|=\mathit{Ga}{l}_K(t^p-u)$ and by by Corollary 6.3.14 we that $\mathit{Ga}{l}_K(t^p-u)|k!⇝\mathit{Ga}{l}_K(t^p-u)|1!$, hence $|\mathit{Gal}(M:K)|=1$. This means that $𝔊=\{\mathit{Gal}(M:K)\}=\{\mathit{id}\}$. Therefore we can clearly see that it is impossible for there to be mutually inverse function between $𝔉$ and $𝔊$.