 For example, the field ℚ(i) of Gaussian rationals, if equipped with the valuation w given by formula_6 (and formula_7) is formally 5-adic (the place "v"=5 of the rationals splits in two places of the Gaussian rationals since formula_8 factors over the residue field with 5 elements, and "w" is one of these places).The field of 5-adic numbers (which contains both the rationals and the Gaussian rationals embedded as per the place "w") is also formally 5-adic.On the other hand, the field of Gaussian rationals is "not" formally 3-adic for any valuation, because the only valuation "w" on it which extends the 3-adic valuation is given by formula_9 and its residue field has 9 elements.