Mathematical Research Letters
Volume 13 (2006)
Abelian extensions of global fields with constant local degree
Pages: 599 – 605
We prove that, given a global field $K$ and a positive integer $n$, there exists an abelian extension $L/K$ (of exponent $n$) such that the local degree of $L/K$ is equal to $n$ at every finite prime of $K$, and is equal to two at the real primes if $n=2$. As a consequence, we prove that the $n$-torsion subgroup of the Brauer group of $K$ is equal to the relative Brauer group of $L/K$.