Galois theory and linear algebra

Let K be a field admitting a Galois extension L of degree n with Galois group G. Artin’s lemma on the independence of characters implies that the algebra of K-linear endomorphisms of L is identical with the set of L-linear combinations of the elements of G. This paper examines some consequences of this description of endomorphisms. We provide a characterization of the rank 1 endomorphisms and describe the matrix-theoretic trace of an endomorphism in terms of the field-theoretic trace. We also investigate in greater detail those endomorphisms annihilating a K-subspace in the case when G is cyclic.