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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.

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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

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).

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