I am trying assignment questions in Field Theory course and couldn't solve completely this particular question.
If u, v and are algebraic over K of degree m and n respectively, then prove that $[K(u, v) K]\leq mn $. If (m, n) =1 , then prove that [K(u, v) :K]= mn.
Using tower rule and if u and v are Linearly independent max possibility of [K(u, v) :K] =mn which will be maximum possibility.
But I am not able to think on how I will proceed when (m, n) =1 . It is equivalent to prove that u and v are independent of each other.
Kindly help me with this case.
We have $$ [K(u,v),K]=[K(u,v),K(u)]\cdot[K(u),K]\\ =[K(u,v),K(u)]\cdot m $$ Which gives us $m\mid [K(u,v),K]$. Similarly we get $n\mid [K(u,v),K]$. If $m$ and $n$ are coprime, this implies $mn\mid [K(u,v),K]$. But since we also have $1\leq [K(u,v),K]\leq mn$, the only possibility is $[K(u,v),K]=mn$.