Recently I have started to study field theory at my own. I am having some confusion in the proof of the primitive element theorem.
Statement of the theorem : $F$ be an infinite field. $K/F$ be a finite field extension. Then $K/F$ is simple iff there exists finitely many intermediate fields of $K/F$.
Proof : $K/F$ is simple extension $\Rightarrow$ $\exists$ $a$ $\in$ $K$ such that $K=F(a)$.
$L$ be any intermediate field, i.e., $F$ $\subset$ $L$ $\subset$ $K$.
$f(x)$ and $g(x)$ be the minimal polynomial of $a$ over $L$ and $F$ respectively. Then $f(x) |g(x)$.
$L'$ be any field generated by $F$ and coefficients of $f(x)$. Then $L'$ $\subset$ $L$ and $f(x)$ is also the minimal polynomial of $a$ over $L'$.
Hence, $[K:L]=deg(f(x))=[K:L']$
Therefore, $[L:L']=1$ $\Rightarrow$ $L=L'$
Up to this part I have understood. I am not being able to understand the next argument that is as follows:
Since $g(x)$ has finitely many distinct monic factors therefore the number of intermediate fields must be finite.
Please help me to understand this. I also could not figure out, where the fact " $L=L'$ " is being used.
Thank you in advance. It will also be a great help if somebody kindly suggest me some books from where, a beginner like me, can do problems of field extension. I am following the book "Fundamentals of Abstract Algebra" by Malik, Mordeson & Sen.
So if $L$ is an intermediate field, then $f(x)|g(x)$. But if $L_2$ is an intermediate field such that $f_2(x)=f(x)$ (where $f_2$ is the minimal polynomial of $a$ over $L_2$), then $L_2=L$ because they both equal to $L'$. So there is an injection from intermediate fields and factors of $g(x)$, so there is only finitely many intermediate field extensions.