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I have learned a lemma that says

For every polynomial $f\in F[x]$ there exists a finite field extension $E\setminus F$ where $f$ is splitted in $E$ and $$[E:F]\leq \deg(f)$$

I wanted to know under what conditions there's an iqualuty $$[E:F]=\deg (f)$$ I guess that a necessary and sufficient condition is that $f$ is irreducible in $F$ and I wanted to ask wheather it's correct.

Thanks.

J. Doe
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  • Actually the correct statement is that if $E$ is a splitting field of $f$ of degree $n$, then $[E:F] \leq n!$ – K. Y Jul 18 '19 at 14:56
  • Moreover, see https://math.stackexchange.com/questions/1149743/biggest-splitting-field-degree-given-a-polynomial-of-degree-n – K. Y Jul 18 '19 at 15:03

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