The picture above is a screenshot from section 5, chapter 14, Artin's Algebra, 1st edition, which is about primitive elements and their applications in Galois theory.
I have difficulty to understand the proof of Proposition (4.4). The proof shown in the book is based on the fact that β is algebraic over F, but what I found most difficult in this proof is exactly to show β is an algebraic element.
(This does sound weird. I guess I might miss some elementary result in the previous chapters and lack some basic knowledge in field theory, since I started reading it from the middle part of the chapter. It could also be because I don't fully understand the Primitive element theorem, which lies right above Proposition (4.4).)
Could anyone help me to understand why β is algebraic over F? Thanks in advance!