How do I prove the statement above? I have noticed my textbook using it but I cannot find an explanation of why that is true. I can see why this is true in things like $Q(\sqrt[4]{2})$ and $Q(\sqrt{2})$ but that is because I can find the polynomial that makes it an extension. I am pretty sure I have to assume $a,c$ are algebraic but i am not sure. Any help would be appreciated.
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2$F$ is a field extension of $E$ iff $E$ is a subfield of $F$. That's all you need. If you want the extension to be algebraic you need to provide more information. – lulu Apr 30 '18 at 00:24
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Just to stress: not every field extension is algebraic. $\mathbb C$ is a field extension of $\mathbb Q$ but it is not an algebraic field extension. – lulu Apr 30 '18 at 00:30
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Yes, because $F(a)\subseteq F(c)$ implies $F(a)(c)=F(c)$. After all, $F(c)\subseteq F(c)(a)=F(a,c)=F(a)(c)$. On the other hand, since $a\in F(c)$ since $F(a)\subseteq F(c)$, so $F(a)(c)=F(a,c)\subseteq F(c)$.
C Monsour
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