Prove F(a,b)=F(a)(b)=F(b)(a) in Field Extensions

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The discussion focuses on proving the equality F(a,b) = F(a)(b) = F(b)(a) in the context of field extensions, specifically when E is an extension of F and a, b are elements of E. The proof involves selecting a basis {u_j} for F(a) over F and a basis {v_k} for F(b) over F. By considering the set {u_jv_k} and employing a dimensional argument, the participants establish the required equality definitively.

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Let E be an extension of F and let a, b belong to E. Prove that F(a,b) =F(a)(b) = F(b)(a).
 
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May we assume that $a$ and $b$ are algebraic over $F$?
 
Deveno said:
May we assume that $a$ and $b$ are algebraic over $F$?

Yes
 
Pick a basis $\{u_j\}$ for $F(a)$ over $F$, and a basis $\{v_k\}$ for $F(b)$ over $F$.

Consider the set: $\{u_jv_k\}$, and use a dimensional argument.
 

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