Abstract
In §3 we have defined a valuation ring of a field K to be a subring A of K such that x ∈ A or x-1 ∈ A for any non-zero x ∈ K. Obviously K is the quotient field of A, and K itself is a valuation ring of K. If K is absolutely algebraic (i.e., algebraic over its prime field) and of prime characteristic, then K is the only valuation ring of K (since any subring of K is a field). We shall see later that all other fields K have valuation rings distinct from K.
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© 1972 Springer-Verlag Berlin Heidelberg
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Endler, O. (1972). Valuation Rings. In: Valuation Theory. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-65505-0_2
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DOI: https://doi.org/10.1007/978-3-642-65505-0_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06070-3
Online ISBN: 978-3-642-65505-0
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