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In mathematics, in particular the study of abstract algebra, a Dedekind–Hasse norm is a function on an integral domain that generalises the notion of a Euclidean function on Euclidean domains. ==Definition== Let ''R'' be an integral domain and ''g'' : ''R'' → Z≥ 0 be a function from ''R'' to the non-negative rational integers. Denote by 0''R'' the additive identity of ''R''. The function ''g'' is called a Dedekind–Hasse norm on ''R'' if the following three conditions are satisfied: * ''g''(0''R'') = 0, * if ''a'' ≠ 0''R'' then ''g''(''a'') > 0, * for any nonzero elements ''a'' and ''b'' in ''R'' either: * * ''b'' divides ''a'' in ''R'', or * * there exist elements ''x'' and ''y'' in ''R'' such that 0 < ''g''(''xa'' − ''yb'') < ''g''(''b''). The third condition is a slight generalisation of condition (EF1) of Euclidean functions, as defined in the Euclidean domain article. If the value of ''x'' can always be taken as 1 then ''g'' will in fact be a Euclidean function and ''R'' will therefore be a Euclidean domain. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dedekind–Hasse norm」の詳細全文を読む スポンサード リンク
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