Let k be a global field, i.e., an algebraic number field or a function field in one variable over a finite field. In this paper the author studies on classifications of finite Galois extensions K/k by its norm group \(N_{K/k}(K^{\times})=\{N_{K/k}(\alpha)|\) \(\alpha \in K^{\times}\}\) in comparison with the local case.
Let K and L be finite Galois extensions of k. Then the author first proves by a cohomological method using class field theory that \(K\subseteq L\) if and only if \(N_{K/k}(K^{\times})\subseteq N_{L/k}(L^{\times})\); in particular, \(K=L\) if and only if \(N_{K/k}(K^{\times})=N_{L/k}(L^{\times})\). Applying this result to decomposable forms, he also proves that there is a subfamily of irreducible (over k) decomposable forms such that \(f(x_ 1,...,x_ n)\) and \(g(x_ 1,...,x_ n)\) are equivalent over k if and only if \(\{f(a_ 1,...,a_ n)|\) \(a_ i\in k\}=\{g(b_ 1,...,b_ n)|\) \(b_ i\in k\}.\)
He next investigates the condition \(N_{F/k}(F^{\times})=N_{K/k}(K^{\times})\cap N_{L/k}(L^{\times})\) for \(F=K \cdot L\supset k\), and defines Property I as follows: A finite Galois extension E/k has Property I, if for any Galois extensions \(k\subseteq K \cdot L\subseteq E\) of k the equality \(N_{F/k}(F^{\times})=N_{K/k}(K^{\times})\cap N_{L/k}(L^{\times})\) holds, where \(F=K\cdot L.\) Under this definition he proves that a finite abelian extension E/k has Property I if and only if the Hasse principle holds for E/k.