If $F$ is a subfield of a field $E$ and if $S = S_1 \cup S_2$, the $F(S) = F(S_1)(S_2)$.
I understand why this has to be so, but I can't seem to prove it rigorously.
Could someone help?
2026-04-30 00:08:30.1777507710
Field extensions and unions
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Notice $S_1\subseteq F(S_1)(S_2)$ and $S_2\subseteq F(S_1)(S_2)$, so $S\subseteq F(S_1)(S_2)$; of course also $F\subseteq F(S_1)(S_2)$ so since $F(S)$ is the smallest subfield of $E$ containing $F$ and $S$ we deduce $F(S)\subseteq F(S_1)(S_2)$.
The reverse inclusion is similar.