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I have the following problem:

$\textbf{(i)}$ Let $K \subset F$ be an algebraic extension that contains splitting field of any polynomial $P\in K[x].$ Prove that $F$ is algebraically closed.

$\textbf{(ii)}$ Let $K \subset F$ be an algebraic extension that contains a root of any polynomial $P\in K[x]\setminus K.$ Prove that $F$ is algebraically closed.

I proved $\textbf{(i)}$ rather easily: We have tower of fields $K\subset F\subset \overline{F}$, where $K\subset F$ and $F\subset \overline{F}$ are algebraic extensions, thus $K\subset \overline{F}$ is also an algebraic extension. For any $\alpha\in \overline{F},$ let us denote its minimal polynomial over $K$ by $m_\alpha\in K[x].$ Since $F$ contains splitting field of $m_\alpha$, in particular we have $\alpha\in F.$ Thus, $\overline{F}\subset F$, i.e. $F=\overline{F}.$

My concern is that in the problem sheet that I am trying to solve the problem $\textbf{(ii)}$ is depicted as something hardcore with lots of hints given. However to me it seems that $\textbf{(ii)}$ is just a reformulation of $\textbf{(i)}$ - if $F$ contains a root $\alpha$ of $P\in K[x]$, then it contains root $\beta$ of $\frac{P}{x-\alpha}\in K[x],$ then it contains root $\gamma$ of $\frac{P}{(x-\alpha)(x-\beta)}\in K[x]$ etc. giving us the whole splitting field of $P$. Is there a flaw in my logic?

Thank you.

Haldot
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    $\frac P{x-\alpha}$ need not be in $K[x]$ because $\alpha$ need not be in $K$. – Andreas Blass Apr 21 '22 at 16:59
  • What is the hint for (ii) ? I would use the primitive element theorem for Galois extensions, then extend to inseparable elements. – reuns Apr 21 '22 at 18:18
  • @AndreasBlass Oh, that's because dividing polynomial $P$ by polynomial $Q$ works like $P=SQ+R$ with $S,R\in F[x]$ only if $P$ and $Q$ were from $F[x],$ right? Thanks a lot, I somehow did not think of it! – Haldot Apr 21 '22 at 23:03
  • The second part is a famous theorem by Gilmer. The proof essentially depends on using that $K$ is perfect (and if not then extending $K$ to a larger but perfect field). See https://math.stackexchange.com/q/3285330/72031 – Paramanand Singh Apr 22 '22 at 01:42

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