On the scaled trace forms and the transfer of a number field extension

Abstract

Let F be a number field. Let L F be a field extension of degree m and let Tr L F be the trace map. In this paper we ask for which quadratic forms ϕ over F with dim ϕ = mn does exist a form ϱ over L with dim ϱ = n such that ϕ is isometric to Tr L F (ϱ) . We further show that for given m, n, r ∈ N and quadratic forms ϕ 1, …, ϕ r over F with dim ϕ i = mn there exist an extension L F of degree m and ϱ 1, …, ϱ r over L with dim ϱ i = n such that Tr L F (ϱ i) is isometric to ϕ i for i = 1, …, r.