Tensor Product of Fields - Compositum of Fields

Compositum of Fields

Firstly, one defines the notion of the compositum of fields. This construction occurs frequently in field theory. The idea behind the compositum is to make the smallest field containing two other fields. In order to formally define the compositum, one must first specify a tower of fields. Let k be a field and L and K be two extensions of k. The compositum, denoted KL is defined to be where the right-hand side denotes the extension generated by K and L. Note that this assumes some field containing both K and L. Either one starts in a situation where such a common over-field is easy to identify (for example if K and L are both subfields of the complex numbers); or one proves a result that allows one to place both K and L (as isomorphic copies) in some large enough field.

In many cases one can identify K.L as a vector space tensor product, taken over the field N that is the intersection of K and L. For example if one adjoins √2 to the rational field ℚ to get K, and √3 to get L, it is true that the field M obtained as K.L inside the complex numbers ℂ is (up to isomorphism)

as a vector space over ℚ. (This type of result can be verified, in general, by using the ramification theory of algebraic number theory.)

Subfields K and L of M are linearly disjoint (over a subfield N) when in this way the natural N-linear map of

to K.L is injective. Naturally enough this isn't always the case, for example when K = L. When the degrees are finite, injective is equivalent here to bijective.

A significant case in the theory of cyclotomic fields is that for the nth roots of unity, for n a composite number, the subfields generated by the pkth roots of unity for prime powers dividing n are linearly disjoint for distinct p.