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

[Scherie]

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).

 

[Deveno]

May we assume that $a$ and $b$ are algebraic over $F$?

 

[Scherie]

May we assume that $a$ and $b$ are algebraic over $F$?

Yes

 

[Deveno]

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.