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The p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems

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Absolutely irreducible p-adic representation of the absolute Galois group of Q_p

One gets loads of crystalline counterexamples using $p$-divisible groups (and more specifically from any elliptic curve with supersingular reduction). Let $k$ be a perfect field of characteristic $ …
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