Galois theory and linear algebra (Q1006028)

This article considers the \(K\)-linear endomorphisms of a (finite) Galois extension \(L\) of a field \(K\). The authors begin by applying Artin's lemma on the independence of characters to show that the set of such endomorphisms is the same as the set of \(L\)-linear combinations of the Galois group \(G = \text{Gal}(L/K)\). This identification is then exploited to prove several additional results. In particular, the rank 1 \(K\)-linear endomorphisms of \(L\) are shown to be precisely those of the form \(\lambda\pi_a := \lambda\sigma_1(a)\sigma_1 + \cdots + \lambda\sigma_n(a)\sigma_n\), where \(\lambda\) and \(a\) are nonzero elements of \(L\), and \(\sigma_1, \dots, \sigma_n\) are the distinct elements of \(G = \text{Gal}(L/K)\). Furthermore, the matrix-theoretic trace of \(\lambda\pi_a\) is shown to be equal to the field-theoretic trace \(\text{Tr}(\lambda a)\). It is shown that any \(K\)-linear endomorphism of \(L\) can be expressed as an \(L\)-linear combination of rank 1 elements of the form \(\pi_a = \sigma_1(a)\sigma_1 + \cdots + \sigma_n(a)\sigma_n\), and this is then used to show that the matrix-theoretic trace of the endomorphism \(\tau = \lambda_1\sigma_1 + \cdots + \lambda_n\sigma_n\) is the field-theoretic trace \(\text{Tr}(\lambda_1\)) The article then focuses on the \(n\times n\) matrix \(B\) whose \(i,j\)-entry is \(\sigma_j(x_i)\), where \(x_1, \dots, x_n\) is a \(K\)-basis for \(L\). A proof is given that the determinant of \(B\) is either in \(K\) or else there is a unique quadratic extension of \(K\) in \(L\), and \(\det(B)\) lies in this quadratic extension. An analogue is obtained for the \((n-1) \times (n-1)\) matrix obtained by using the nonidentity elements of \(G\) and a basis for the trace zero hyperplane \(H_0 = \ker(\text{Tr}_{L/K}\)). The article concludes with some results specific to the case when \(G\) is a cyclic group, providing an explicit description of the \(K\)-linear endomorphisms of \(L\) annihilating a \(K\) subspace \(U\) of \(L\).