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Let $E_i (i=1, 2)$ be subfields of $K$ containing the subfield $F$, and $[E_i : F] < \infty$. If $E$ is the subfield of $K$ generated by $E_1$ and $E_2$, then $[E:F] \leqslant [E_1 : F][E_2 : F]$.

Could you give me some hint? In fact, according to $[E:F] = [E : E_1][E_1 : F]$, I know we need to prove $[E:E_1]\leqslant [E_2: F]$, but what should I do next?

Temoi
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Klein
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  • Do you mean subfield? – Temoi Jan 15 '24 at 14:42
  • @MatteoIaccarino, of course sir. – Klein Jan 15 '24 at 14:50
  • @MatteoIaccarino Not entirely, I did see his approach, but I prefer to tackle it through the line of questioning that I proposed. – Klein Jan 15 '24 at 15:40
  • Please edit your post to replace "subdomain" by "subfield" everywhere, and "$E$ is a..." by "$E$ is the...", and choose a more descriptive title. – Anne Bauval Jan 15 '24 at 17:19
  • Actually, [the accepted answer to] the post linked by Matteo does answer your question: instead of writing "${a_ib_j\mid 1\le i\le n, 1\le j\le m}$ is a spanning set" (of $E$ over $F$), simply write : "${b_j\mid1\le j\le m}$ is a spanning set of $E$ over $E_1$", but don't change any word in the proof. – Anne Bauval Jan 15 '24 at 17:35

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