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Let $K$ and $L$ be extensions of a finite field $F$ of degrees $n$ and $m$,respectively.

How do I show that $KL$ has degree $\mathrm{lcm}(m,n)$ over $F$ and $K\cap F$ has degree $\gcd(m,n)$ over $F$?

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Don't yell! – Arturo Magidin Mar 25 '12 at 3:46
Since you've done it twice now: all capital letters, on the internet, is the equivalent of shouting. Rather a rude way of trying to call attention, don't you think? – Arturo Magidin Mar 25 '12 at 4:09

Hint. A field of order $p^n$ is contained in a field of order $p^m$ if and only if $n|m$. $KL$ is the smallest field that contains both $K$ and $L$, and $K\cap F$ is the largest field contained in both $K$ and $L$. Translate the inclusion statements into divisibility statements.

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