Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $M$ be a finite extension of $E$ and let $E$ be a finite extension of $F$. Then $M$ is a finite extension of $F$ and $[M:F] = [M:E][E:F]$.

Is there an easy explanation and/or proof for this theorem? My instructor gave an incomplete proof in class but got stuck and I'm kind of confused of what's going on.

share|improve this question
8  
I would be surprised if you could pick a textbook on field theory which does not prove this in detail! –  Mariano Suárez-Alvarez Nov 23 '11 at 5:40
add comment

1 Answer

Field extensions are vector spaces for their ground fields: If you have an $F$-basis of $E$, say $(e_1,...,e_n)$ and an $E$-basis of $M$, $(m_1,...,m_k)$, then $$\mathcal{B}:=(e_1 \cdot m_1, e_1 \cdot m_2, ... , e_1 m_k, e_2 m_1, ... , e_n m_k)$$ is an $F$-basis of $M$. To prove this check that $\mathcal{B}$ generates $M$ and that $\mathcal{B}$ is linearly independent (using the fact that the two smaller bases generate $M$ and $E$ and that those bases are linearly independent).

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.