On an Obstruction to the Hasse Norm Principle and the Equality of Norm Groups of Algebraic Number Fields

Abstract

Let LÂk and TÂk be finite extensions of algebraic number fields. In the present work we introduce the factor group ofwhere J L and J T are the idele groups of L and T, respectively. The main theorem shows that the computation of this factor group can be reduced to the computation in finite group theory, and the computation with Galois groups of local extensions at a finite number of primes of the base field k. We then apply the main theorem to establish a number of interesting results on the equality of norm groups as subgroups of the multiplicative group of k. In particular, we obtain new results on solitary non-Galois extensions of algebraic number fields.

On the norm groups of Galois 2n-extensions of algebraic number fields

“…Then τ j ∈ N G 1 (M 1 ) for each 1 j f , and τ ∈ N G 2 (M 2 ). Indeed, by (11) and (13) for any h ∈ M 1 and 1 j f we obtain (τ −1 (1). So τ −1 j hτ j ∈ M 1 , and therefore τ j ∈ N G 1 (M 1 ) for each 1 j f .…”

“…By Theorem (1.11) of [11, p. 252] [11]. We will use the group theoretic interpretation (2) to prove that F i /P satisfies HNP for each i = 1, 2.…”

“…Let L/k and T /k be finite extensions of an algebraic number field k. The factor group of k * ∩ N L/k I L N T /k I T by N(T /k)N L/k L * is called the first obstruction to HNP for L/k corresponding to T /k [11]. In the special case when T is a Galois extension of k containing L the first obstruction is of the form N(L/k)/N(T /k)N L/k L * .…”

“…In [11] we introduced the first obstruction to HNP as follows. Let L/k and T /k be finite extensions of an algebraic number field k. The factor group of k * ∩ N L/k I L N T /k I T by N(T /k)N L/k L * is called the first obstruction to HNP for L/k corresponding to T /k [11].…”