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We show that Hermite's approximations to values of the exponential functionat given algebraic numbers are nearly optimal when considered from an adelicperspective. We achieve this by taking into account the ratio of these valueswhenever they make sense in the various completions (Archimedean or $p$-adic)of a number field containing these algebraic numbers.

Approximation simultan\'ee des valeurs de la fonction exponentielle dans les ad\`eles