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In algebra, a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has Pythagoras number equal to 1. A Pythagorean extension of a field ''F'' is an extension obtained by adjoining an element for some λ in ''F''. So a Pythagorean field is one closed under taking Pythagorean extensions. For any field ''F'' there is a minimal Pythagorean field ''F''py containing it, unique up to isomorphism, called its Pythagorean closure.〔Milnor & Husemoller (1973) p. 71〕 The ''Hilbert field'' is the minimal ordered Pythagorean field.〔Greenberg (2010)〕 ==Properties== Every Euclidean field (an ordered field in which all positive elements are squares) is an ordered Pythagorean field, but the converse does not hold.〔Martin (1998) p. 89〕 A quadratically closed field is Pythagorean field but not conversely (R is Pythagorean); however, a non formally real Pythagorean field is quadratically closed.〔Rajwade (1993) p.230〕 The Witt ring of a Pythagorean field is of order 2 if the field is not formally real, and torsion-free otherwise.〔 For a field ''F'' there is an exact sequence involving the Witt rings : where ''I'' ''W''(''F'') is the fundamental ideal of the Witt ring of ''F''〔Milnor & Husemoller (1973) p. 66〕 and Tor ''I'' ''W''(''F'') denotes its torsion subgroup (which is just the nilradical of ''W''(''F'').〔Milnor & Husemoller (1973) p. 72〕 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「Pythagorean field」の詳細全文を読む スポンサード リンク
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