Artin decomposition for p-adic numbers

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You can use the .slice() method. For the frac function, you may have to decide wether you want to keep the precision of your adic number or if you want to reset it to 1:

sage: floor_adic = lambda a : a.slice(0, a.precision_absolute())
sage: frac_adic = lambda a : a - a.slice(0, a.precision_absolute())
sage: frac_adic_noprec = lambda a : a.slice(a.valuation(), 0)

sage: K = Qp(11); K
11-adic Field with capped relative precision 20
sage: b = K(3211/11^2); b
10*11^-2 + 5*11^-1 + 4 + 2*11 + O(11^18)

sage: floor_adic(b)
4 + 2*11 + O(11^18)
sage: frac_adic(b)
10*11^-2 + 5*11^-1 + O(11^18)
sage: frac_adic_noprec(b)
10*11^-2 + 5*11^-1 + O(11^0)