Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 18H / b / Solution

The products $xy$ with $x\in L_1$ and $y\in L_2$ span the field compositum $L_1{L}_2$ over $K$, soThe tower law givesThus the degree is divisible by both $d_1$ and $d_2$. Since they are coprime, it is divisible by $d_1{d}_2$. Combining divisibility with the upper bound givesEquivalently, finite extensions of coprime degrees are linearly disjoint field extensions.