Degree of A Field Extension - The Multiplicativity Formula For Degrees

The Multiplicativity Formula For Degrees

Given three fields arranged in a tower, say K a subfield of L which is in turn a subfield of M, there is a simple relation between the degrees of the three extensions L/K, M/L and M/K:

In other words, the degree going from the "bottom" to the "top" field is just the product of the degrees going from the "bottom" to the "middle" and then from the "middle" to the "top". It is quite analogous to Lagrange's theorem in group theory, which relates the order of a group to the order and index of a subgroup — indeed Galois theory shows that this analogy is more than just a coincidence.

The formula holds for both finite and infinite degree extensions. In the infinite case, the product is interpreted in the sense of products of cardinal numbers. In particular, this means that if M/K is finite, then both M/L and L/K are finite.

If M/K is finite, then the formula imposes strong restrictions on the kinds of fields that can occur between M and K, via simple arithmetical considerations. For example, if the degree is a prime number p, then for any intermediate field L, one of two things can happen: either = p and = 1, in which case L is equal to K, or = 1 and = p, in which case L is equal to M. Therefore there are no intermediate fields (apart from M and K themselves).