The concept plays an important role in many parts of mathematics, including algebra and number theory—indeed in any area where fields appear prominently.
The dimension of this vector space is called the degree of the field extension, and it is denoted by [E:F].
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.
If x is any element of M, then since the wn form a basis for M over L, we can find elements an in L such that Then, since the um form a basis for L over K, we can find elements bm,n in K such that for each n, Then using the distributive law and associativity of multiplication in M we have which shows that x is a linear combination of the umwn with coefficients from K; in other words they span M over K. Secondly we must check that they are linearly independent over K. So assume that for some coefficients bm,n in K. Using distributivity and associativity again, we can group the terms as and we see that the terms in parentheses must be zero, because they are elements of L, and the wn are linearly independent over L. That is, for each n. Then, since the bm,n coefficients are in K, and the um are linearly independent over K, we must have that bm,n = 0 for all m and all n. This shows that the elements umwn are linearly independent over K. This concludes the proof.
Both dimensions however satisfy a multiplication formula for towers of division rings; the proof above applies to left-acting scalars without change.