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A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), the point can be constructed with unruled straightedge and compass. A complex number is a constructible number if its corresponding point in the Euclidean plane is constructible from the usual ''x''- and ''y''-coordinate axes. It can then be shown that a real number ''r'' is constructible if and only if, given a line segment of unit length, a line segment of length |''r'' | can be constructed with compass and straightedge.〔John A. Beachy, William D. Blair; ''Abstract Algebra''; (Definition 6.3.1 )〕 It can also be shown that a complex number is constructible if and only if its real and imaginary parts are constructible. In terms of algebra, a number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots. The set of constructible numbers can be completely characterized in the language of field theory: the constructible numbers form the quadratic closure of the rational numbers: the smallest field extension that is closed under square root and complex conjugation. This has the effect of transforming geometric questions about compass and straightedge constructions into algebra. This transformation leads to the solutions of many famous mathematical problems, which defied centuries of attack. == Geometric definitions == The geometric definition of a constructible point is as follows. First, for any two distinct points ''P'' and ''Q'' in the plane, let ''L''(''P'', ''Q'' ) denote the unique line through ''P'' and ''Q'', and let ''C'' (''P'', ''Q'' ) denote the unique circle with center ''P'', passing through ''Q''. (Note that the order of ''P'' and ''Q'' matters for the circle.) By convention, ''L''(''P'', ''P'' ) = ''C'' (''P'', ''P'' ) = . Then a point ''Z'' is ''constructible from E, F, G and H'' if either #''Z'' is in the intersection of ''L''(''E'', ''F'' ) and ''L''(''G'', ''H'' ), where ''L''(''E'', ''F'' ) ≠ ''L''(''G'', ''H'' ); #''Z'' is in the intersection of ''C'' (''E'', ''F'' ) and ''C'' (''G'', ''H'' ), where ''C'' (''E'', ''F'' ) ≠ ''C'' (''G'', ''H'' ); #''Z'' is in the intersection of ''L''(''E'', ''F'' ) and ''C'' (''G'', ''H'' ). Since the order of ''E'', ''F'', ''G'', and ''H'' in the above definition is irrelevant, the four letters may be permuted in any way. Put simply, ''Z'' is constructible from ''E'', ''F'', ''G'' and ''H'' if it lies in the intersection of any two distinct lines, or of any two distinct circles, or of a line and a circle, where these lines and/or circles can be determined by ''E'', ''F'', ''G'', and ''H'', in the above sense. Now, let ''A'' and ''A''′ be any two distinct fixed points in the plane. A point ''Z'' is ''constructible'' if either #''Z'' = ''A''; #''Z'' = ''A''′; #there exist points ''P''0, ..., ''P''''n'', with ''Z'' = ''P''''n'', such that for all ''j'' ≥ 1, ''P''''j'' + 1 is constructible from points in the set where ''P''0 = ''A'' and ''P''1 = ''A''′. Put simply, ''Z'' is constructible if it is either ''A'' or ''A''′, or if it is obtainable from a finite sequence of points starting with ''A'' and ''A''′, where each new point is constructible from previous points in the sequence. For example, the center point of ''A'' and ''A''′ is defined as follows. The circles ''C'' (''A'', ''A''′) and ''C'' (''A''′, ''A'') intersect in two distinct points; these points determine a unique line, and the center is defined to be the intersection of this line with ''L''(''A'', ''A''′). 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「constructible number」の詳細全文を読む スポンサード リンク
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