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help me with this excercice..

Let $F$ be a field and $K$ a splitting field for some nonconstant polynomial over $F$. Show that $K$ is a finite extension of $F$.

I try

$K$ is a splitting field.

$K=F(a_1,a_2,...,a_n)$ for $a_i$ roots for polynomial over $F$, i guess that have proof $[K:F]=n$ help

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    $\begingroup$ Hint: Try to prove that for all $i$ the extension $F(a_1,a_2,\ldots,a_i)/F(a_1,a_2,\ldots,a_{i-1})$ is finite. Then apply what you know about towers of extensions. $\endgroup$
    – Jyrki Lahtonen
    Dec 2, 2016 at 5:26
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    $\begingroup$ Note that it's not generally true that $[K:F]=n$. Indeed, $[K:F]$ can be as small as 1, or as big as $n$-factorial. $\endgroup$
    – Gerry Myerson
    Dec 2, 2016 at 6:28

1 Answer 1

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Hint:

First prove it for one root and use induction for $n$ roots. Suppose $K=F(a)$ and $|K:F|=m$. Then for any $x\in K$, $1, x, \cdots, x^m$ must be linear dependent because $K$ is vector space with dimension of $m$.

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  • $\begingroup$ mm thanks,,I think to have the proof complete :) $\endgroup$
    – Yobamath
    Dec 2, 2016 at 7:09
  • $\begingroup$ If you have a proof now, Yoba, let me encourage you to write it up and post it as answer to your question. Also, you have the option of "accepting" 1006's answer by clicking in the check mark box next to it. $\endgroup$
    – Gerry Myerson
    Dec 2, 2016 at 8:41

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