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- W34761469 abstract "Let F be an algebraic number field. The set of valuations on F is denoted by M F . Let |.| v , or v in shorthand, be a valuation of F. Denote by F v the completion of F with respect to v. If F v is ℝ or ℂ we assume that v coincides with the usual absolute value on these fields. When v is a finite valuation we assume it normalised by |p| v = 1/p where p is the unique rational prime such that |p| v < 1. The normalised valuation ‖.‖ v is defined by $$ left| x right|_v = left| x right|_v^{[F_v :mathbb{Q}_p ]/[F:mathbb{Q}]} $$with the convention that p = ∞ when v is archimedean and ℚ∞ = ℝ. For any non-zero x ∈ F we have the product formula$$ prodlimits_v {left| x right|_v = 1} . $$(1.1)" @default.
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- W34761469 date "2009-02-04" @default.
- W34761469 modified "2023-09-23" @default.
- W34761469 title "Diophantine Equations and Approximation" @default.
- W34761469 doi "https://doi.org/10.1007/978-3-540-48208-6_1" @default.
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