Abstract
We embed the important work of Gräter on approximation theorems in the book. Approximation theorems are a well-known and important topic in classical valuation theory of fields. The question is to decide for given valuations v 1, …, v n of a field, elements a 1, …, a n in the field and α 1, …, α n in the value groups whether there is an element x in the field such that
for all i; i.e. if the elements a i can be approximated by some x up to a certain degree. Gräter elaborated various approximation theorems in our general setting of R-Prüfer rings and has found deep connections, to be reflected below.
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Notes
- 1.
This means that f(α) ≥ f(β) if α ≥ β (cf. [Vol. I, p. 17]). Note that necessarily \(f(\varGamma _{v}) \subset \varGamma _{w}\) and that \(f\vert \varGamma _{v}:\varGamma _{v} \rightarrow \varGamma _{w}\) is a homomorphism of ordered groups.
- 2.
Note that then necessarily f(∞) = ∞.
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Knebusch, M., Kaiser, T. (2014). Approximation Theorems. In: Manis Valuations and Prüfer Extensions II. Lecture Notes in Mathematics, vol 2103. Springer, Cham. https://doi.org/10.1007/978-3-319-03212-2_2
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DOI: https://doi.org/10.1007/978-3-319-03212-2_2
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