Here is the proof of Lemma A-5.19 in Chapter A-5 of Rotman's book Advanced Modern Algebra. (What he calls a 'pure extension' is commonly called 'radical extension' by most authors.)
I am confused by the case where $\mathrm{Characteristic}\,k = p_{1}$, as underlined in the screenshot above. It is based on a former example, which is as following:
In Example A-5.8, the base field $k$ is assumed to be in the form $\mathbb{F}_{p}(t)$ where $\mathbb{F}_{p}$ is a finite field of order $p$. But in Lemma A-5.19 this assumption is dropped, and the base field is only assumed to be $\mathrm{Characteristic}\,k$. Can anyone help me understand why the assumption can be dropped while using Example A-5.8 to prove Lemma A-5.19?