Galois Descent for Vector Spaces
Let \(E/k\) be a finite Galois extension with group \(\Gamma = \operatorname{Gal}(E/k)\), and let \(V\) be an \(E\)-vector space with a semilinear \(\Gamma\)-action: additive automorphisms \(\rho_\gamma\) with \(\rho_\gamma(c v) = \gamma(c)\,\rho_\gamma(v)\), compatible with composition. Then the natural map \[ E \otimes_k V^{\Gamma} \;\longrightarrow\; V \] is an isomorphism of \(E\)-vector spaces. (Classically Speiser's lemma; the case \(V = E^n\) with the standard action is equivalent to \(H^1(\Gamma, \mathrm{GL}_n(E)) = 1\), the Hilbert 90 statement for the general linear group.)
Equivalently, split into the two halves that are used separately:
- Spanning (surjectivity): the fixed points \(V^{\Gamma}\) span \(V\) over \(E\). This half does not use that \(E/k\) is Galois — only that \(E/k\) is finite, so that \(\Gamma\) is a finite set of field embeddings.
- Independence (injectivity): \(k\)-linearly independent elements of \(V^{\Gamma}\) remain \(E\)-linearly independent in \(V\).
Consequently the category of \(k\)-vector spaces is equivalent to the category of \(E\)-vector spaces with semilinear \(\Gamma\)-action, \(W \mapsto E \otimes_k W\) with quasi-inverse \(V \mapsto V^\Gamma\). This equivalence is the engine behind the classification of twisted forms — in particular the classification of algebraic tori by Galois lattices (F-Tori Are Classified by Galois Modules), where it is applied to the group algebra \(E[M]\) with the semilinear action twisted by a lattice action.
Proof
Spanning. For \(v \in V\) and \(c \in E\), the average \(w_c = \sum_{\gamma \in \Gamma} \rho_\gamma(c v)\) is \(\Gamma\)-fixed, and by semilinearity \(w_c = \sum_\gamma \gamma(c)\, \rho_\gamma(v)\). The evaluation vectors \((\gamma(c))_{\gamma \in \Gamma}\), as \(c\) ranges over \(E\), span the function space \(\Gamma \to E\) over \(E\): a linear functional annihilating all of them is a relation \(\sum_\gamma a_\gamma \gamma = 0\) among the distinct field embeddings, which forces \(a_\gamma = 0\) by Dedekind's linear independence of characters. Choose therefore finitely many \(c_i \in E\) and \(b_i \in E\) with \(\sum_i b_i \gamma(c_i) = \delta_{\gamma, 1}\) for all \(\gamma \in \Gamma\); then \(\sum_i b_i\, w_{c_i} = \sum_\gamma \bigl(\sum_i b_i \gamma(c_i)\bigr) \rho_\gamma(v) = \rho_1(v) = v\), exhibiting \(v\) in the \(E\)-span of fixed vectors.
Independence. Let \(v_1, \dots, v_n \in V^\Gamma\) be \(k\)-linearly independent and suppose \(\sum_j c_j v_j = 0\) with \(c_j \in E\), not all zero, with \(n\) minimal. Normalizing \(c_1 = 1\) and applying \(\rho_\gamma\) gives \(\sum_j \gamma(c_j) v_j = 0\); subtracting, \(\sum_{j \geq 2} (\gamma(c_j) - c_j) v_j = 0\) is a shorter relation, so \(\gamma(c_j) = c_j\) for all \(\gamma\), i.e. all \(c_j \in E^\Gamma = k\) (here Galois-ness of \(E/k\) enters), contradicting \(k\)-independence.
Isomorphism. Surjectivity of \(E \otimes_k V^\Gamma \to V\) is the spanning half; injectivity is the independence half applied to a \(k\)-basis of \(V^\Gamma\).