Exercise 5.1.20 (Prime degree is simple)

Solution: Let $M:K(\alpha ):K$ be field extensions where $M$ and $K$ are arbitrary fields, $\alpha \in M$, and $[M:K]=p$, where $p$ is prime. By Theorem 5.1.17 (iii) we have $[M:K]=[M:K(\alpha )][K(\alpha ):K]$. Hence, we must have that $[K(\alpha ):K]=1$ or $p$, however, we also know that $K(\alpha )\ne K$, hence $[K(\alpha ):K]=p$, and therefore, $[M:K(\alpha )]=1$, which by Example 5.1.3 tells us $M=K(\alpha )$. Hence $M:K$ is a simple.