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The Artin-Whaples approximation theorem October 6, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, number theory.
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The Artin-Whaples approximation theorem is a nice extension of the Chinese remainder theorem to absolute values, to which it reduces when the absolute values are discrete.

So fix pairwise nonequivalent absolute values {\left|\cdot\right|_1, \dots, \left|\cdot\right|_n} on the field {K}; this means that they induce different topologies, so are not powers of each other

Theorem 1 (Artin-Whaples)

Hypotheses as above, given {a_1, \dots, a_n \in K} and {\epsilon>0}, there exists {a \in K} with

 

\displaystyle \left|a - a_i\right|_i < \epsilon, \quad 1 \leq i \leq n.

  (more…)