0
$\begingroup$

Let $F$ be a global field with integers $o$, and let $x \in F$. Does $|x|_v =1$ for all non-archimedean valuations of $F$ imply that $x \in o^\times$.

1 Answers 1

2

Let $x\cal O$ the (possibly fractional) ideal generated by $x$. If it is non-trivial, i.e. $\cal O\neq x\cal O$, it must be a product of (possibly some inverted) prime ideals. The valuation of $x$ at those primes is not $1$.

Thus.....

  • 0
    I do not understand. Do you say $|x|_v \neq 1$ does not imply $x \in o^\times$? Do you assume that $o$ is the ring of integers of $F$, not $o \neq \mathbb{Z}$?2012-07-19
  • 1
    What I'm saying is that an element which is a local unit at all primes must generate the trivial fractional ideal, but then it has to be a global unit.2012-07-19
  • 0
    Okay, so in short the answer is yes. Thanks.2012-07-19