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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 $x \in o^\times$.

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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.....

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    Okay, so in short the answer is yes. Thanks.2012-07-19