degree of an extension

An extension field L sitting on top of a smaller field K can be viewed as a vector space: you can add elements of L and scale them by numbers from K. The degree of the extension just asks how many directions this vector space has — its dimension. A degree of 2 means every element of L is captured by two coordinates over K; a degree of 6 means six. It is a single number measuring how much bigger L is than K.

Formally, the degree [L : K] is the dimension of L as a K-vector space, dim_K(L). When this dimension is finite the extension is called finite of that degree; when it is infinite the extension is infinite. For a simple algebraic extension K(a), the degree equals the degree of the minimal polynomial of a over K, so adjoining a root of an irreducible cubic gives degree 3.

The degree controls almost everything downstream: it is finite exactly when L is a finite-dimensional, hence algebraic, extension; it multiplies through towers; and for Galois extensions it equals the order of the Galois group. A transcendental extension always has infinite degree, since adjoining a free variable already produces the infinite-dimensional space K(t).