tower law

Suppose you build a field upward in two stages: from K to an intermediate field M, then from M to a larger field L. The tower law says the total degree is simply the product of the two step degrees. If the first step doubles the size and the second triples it, the whole climb multiplies it by six. Stacking field extensions behaves exactly like multiplying dimensions, which matches your intuition that doing two enlargements in sequence compounds them.

Formally, for fields K subset of M subset of L, [L : K] = [L : M] times [M : K], including the case where either factor is infinite. The proof is concrete: if {a_i} is an M-basis of L and {b_j} is a K-basis of M, then the products {a_i b_j} form a K-basis of L, so the dimension counts multiply exactly.

This single identity is one of the most-used tools in field theory. It forces any intermediate field M of a degree-[L:K] extension to have degree dividing [L : K], which immediately rules out, say, a subfield of degree 2 inside a degree-3 extension. It is the engine behind impossibility proofs for compass-and-straightedge constructions and behind much of Galois theory.