Theorem-1 Let $f(x)\in F[x]$ be any polynomial of degree $n\geq 1$, then no extension of $F$ contain more than $n$ roots of $f(x)$.
Theorem-2 Let $f(x)\in F[x]$ be any polynomial of degree $n$ then there exist an extension $E$ of $F$ containing all the roots of $f(x)$ and $[E:F]\leq n!$.
I know the proof of $1^{st}$ theorem. I believe I can proof Theorem-2.
I wanna know that what is the difference between both theorems. I think in Theorem-2 we have splitting filed of polynomial $f(x)$ which makes difference from theorem-1.
Theorem-1 states that for a given polynomial of degree $n\geq 1$ on a field $F$ ($p\in F[X]$) there are no field extention $E/ F$ in which there exists more than $n$ roots. We are talking about the polynomial $p$ and how it cannot contains more than $n$ roots whichever the field we look it on.
Seems pretty intuitive, we can look at a polynomial in $\mathbb{C}$ to comfort ourselves: $$p(z)=a_n z^n+\cdots+a_1 z + a_0$$ We see that $p$ cannot have more than $n$ roots otherwise we get this: $R:=\{\alpha\in\mathbb{C}\mid p(\alpha)=0\}$ and say that $|R|= n+1$, we could then factor linear functions out of $p$ for $n+1$ times. But the $\deg p\neq n$
Theorem-2 talks about the existence of an extension in which we can split the polynomial and the degree of such an extension. When we say split it's refering to the factorisation of the polynomial but it isn't so far of saying that the roots are contained in the extension. To get more information on it, you can take a look at Splitting field which is what will usually follow in the course/book.
So to clarify: