Galois Descent

Let L/K be a field extension. A K-vector space W can be extended to an L-vector space L⊗KW , and W embeds into L⊗KW by w 7→ 1⊗w. Under this embedding, when W 6= 0 a K-basis {ei} of W turns into an L-basis {1⊗ ei} of L⊗KW . Passing from W to L⊗KW is called ascent. In the other direction, if we are given an L-vector space V 6= 0, we may ask how to describe the K-subspaces W ⊂ V such that a K-basis of W is an L-basis of V . Such a K-subspace W is called a K-form of V . For completeness, when V = 0 (so there is no basis), we regard W = 0 as a K-form of V . The passage from an L-vector space V to a K-form of V is called descent. Whether we can descend is the question of filling in the question mark in the figure below.