Given three fields, the tower law states that if and are both finite then so is and And furthermore if either or is infinite then so is .
First consider the case where and are both finite. Let and . Let be a basis for over and be a basis for over . We claim that the set (which clearly has elements) is a basis for over .
First we show that spans . Take any . As is a basis for over , we can write , where . And now as is a basis for over we can write where , for each . So now So indeed spans over .
Now we show that is independent. Assume that there are some such that . So then we have So, as is independent over we get that For all . And hence as is independent over we get for all and . Therefore is indeed independent.
Therefore is indeed a basis, so , as desired.
Now we consider the infinite case. By the above argument if is independent over and is independent over then the set is independent over . Hence if either of and is infinite then there exisit arbitrarily large independent sets in over , so is infinite as well.