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In number theory, a rational reciprocity law is a reciprocity law involving residue symbols that are related by a factor of +1 or –1 rather than a general root of unity. As an example, there are rational biquadratic and octic reciprocity laws. Define the symbol (''x''|''p'')''k'' to be +1 if ''x'' is a ''k''-th power modulo the prime ''p'' and -1 otherwise. Let ''p'' and ''q'' be distinct primes congruent to 1 modulo 4, such that (''p''|''q'')2 = (''q''|''p'')2 = +1. Let ''p'' = ''a''2 + ''b''2 and ''q'' = ''A''2 + ''B''2 with ''aA'' odd. Then : If in addition ''p'' and ''q'' are congruent to 1 modulo 8, let ''p'' = ''c''2 + 2''d''2 and ''q'' = ''C''2 + 2''D''2. Then : ==References== * * * * 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「rational reciprocity law」の詳細全文を読む スポンサード リンク
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