 Tower law

For finite field extensions $K\subseteq L\subseteq M$, the degree of a field extension satisfiesThis tower law is a statement about the dimension of the underlying vector spaces.

Choose a basis $u_1,\dots ,u_r$ of $L$ over $K$ and a basis $v_1,\dots ,v_s$ of $M$ over $L$. The products $u_i{v}_j$ span $M$ over $K$: first expand an element using the $v_j$, then expand each coefficient using the $u_i$.

They are also linearly independent. If ${\sum }_{i,j}{a}_{ij}{u}_i{v}_j=0$ with $a_{ij}\in K$, independence of the $v_j$ gives ${\sum }_i{a}_{ij}{u}_i=0$ for every $j$; independence of the $u_i$ then gives every $a_{ij}=0$. There are $rs$ products, proving the formula.