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What is an example of a better way to write a proof than the one I have written?

My question is: What is an example of a better way to write a proof than the one I have written below?
Suppose $F$ is a real field and $a$ is an element of $F$ which is algebraic over $F$. We want to show that $a$ is a root of the minimal polynomial $p(x)$ over $F$.
Let $E$ be the splitting field of $p(x)$ over $F$ and let $m(x) \in F[x]$ be the minimal polynomial of $a$ over $F$. We show the following:
1) If $x$ is a root of $p(x)$ then $m(x)$ is also a root of $p(x)$.
2) If $g(x) \in F[x]$ is another polynomial which is the minimal polynomial of $a$ over $F$ and $x$ is a root of $g(x)$ then $m(x)$ is also a root of $g(x)$.
3) It suffices to show that $p(x) \mid m(x)$.
4) Suppose that $m(x)$ has degree $n$. If $a_1, \dots, a_n \in F$ are distinct solutions to $m(x) = 0$, then $a_1, \dots, a_n$ are distinct roots of $p(x)$.

A:

Are you reading Artin or W.S. Adams?
It would be more natural to phrase the proof like this:
Suppose $x$ is a root of $p(x)$. By the assumption $p(x)$ splits over $F$, so $m(x)$ is the product of the distinct irreducible factors of $p(x)$.
This implies that if $a_1, \ldots, a_n$ are the distinct roots of $m(x)$, then $a_