# extension of a field

• Apr 18th 2010, 07:26 PM
apple2009
extension of a field
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.
• Apr 18th 2010, 07:44 PM
tonio
Quote:

Originally Posted by apple2009
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 $F_1\nsubseteq F_2$ $\,,\,F_2\nsubseteq F_1\Longrightarrow \exists f_1\in F_1-F_2\,,\,\exists f_2\in F_2-F_1$ . Now the question is: where does $f_1+f_2$ belong to?

Tonio