Hahn-Witt Series
The construction of Hahn series can be combined with Witt vectors (at least over a perfect field) to form “twisted Hahn series” or “Hahn-Witt series”: for example, over a finite field K of characteristic p (or their algebraic closure), the field of Hahn-Witt series with value group Γ (containing the integers) would be the set of formal sums where now are Teichmüller representatives (of the elements of K) which are multiplied and added in the same way as in the case of ordinary Witt vectors (which is obtained when Γ is the group of integers). When Γ is the group of rationals or reals and K is the algebraic closure of the finite field with p elements, this construction gives a (ultra)metrically complete algebraically closed field containing the p-adics, hence a more or less explicit description of the field or its spherical completion.
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