Corollary

Let L be a field. Let Γ be a subgroup of the automorphism group of the field L, and let K be the field of invariants of Γ. Let V be a right K-linear subspace of L of finite dimension $n$ over K. Then there exist elements $\sigma_1, . . . , \sigma_n$ of Γ with the following property: for every K-linear mapping $u$ from V to L, there exist elements $a_1, . . . , a_n$ of L such that we have $u(x) =\sum_{i=1}^na_i\sigma_i(x)$ for every $x$ in V.

Denote by E the set of mappings from L to L of the form $x\mapsto$ $\sum_{\sigma\in\Gamma}a_{\sigma}\sigma (x)$, where $(a_{\sigma})_{\sigma\in\Gamma}$ is a family of elements of L with finite support. We have $\boldsymbol{\gamma }_a\in E$ for every $a$ in L, and E is a subring of the endomorphism ring of the additive group of L. Moreover, the field K consists of the elements $a$ of L such that $u(xa) =u(x)a$ for every $x$ in L and every $u$ in E.

Let H be the left L-vector space Hom$_K(V,L)$; it has dimension $n$. By Proposition 9, it is generated by the restrictions of the elements of Γ to V. There exist $n$ elements $\sigma_1, . . . , \sigma_n$ of Γ whose restrictions to V form a basis of H over L. The corollary follows from this.