Prove: Let E be a finite extension of a field F. Let F₁ and F₂ be subfields of E containing F. Then F₁+F₂ is not a subfield of E, except in the cases when F₁ is subset of F₂ or F₂ is subset of F₁.
Prove: Let E be a finite extension of a field F. Let F₁ and F₂ be subfields of E containing F. Then F₁+F₂ is not a subfield of E, except in the cases when F₁ is subset of F₂ or F₂ is subset of F₁.
Suppose $\displaystyle F_1\nsubseteq F_2$ $\displaystyle \,,\,F_2\nsubseteq F_1\Longrightarrow \exists f_1\in F_1-F_2\,,\,\exists f_2\in F_2-F_1$ . Now the question is: where does $\displaystyle f_1+f_2$ belong to?