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The Mason–Stothers theorem, or simply Mason's theorem, is a mathematical theorem about polynomials, analogous to the abc conjecture for integers. It is named after W. Wilson Stothers, who published it in 1981,〔.〕 and R. C. Mason, who rediscovered it shortly thereafter.〔.〕 The theorem states: :Let , , and be relatively prime polynomials over a field such that and such that not all of them have vanishing derivative. Then :: Here is the product of the distinct irreducible factors of ''f''. For algebraically closed fields it is the polynomial of minimum degree that has the same roots as ; in this case gives the number of distinct roots of . ==Examples== *Over fields of characteristic 0 the condition that ''a'', ''b'', ''c'' do not all have vanishing derivative is equivalent to the condition that they are not all constant. Over fields of characteristic it is not enough to assume that they are not all constant. For example, the identity gives an example where the maximum degree of the three polynomials ( and as the summands on the left hand side, and as the right hand side) is , but the degree of the radical is only . *Taking ''a''(''t'')=''t''''n'' and ''c''(''t'') = (''t''+1)''n'' gives an example where equality holds in the Mason–Stothers theorem, showing that the inequality is in some sense the best possible. *A corollary of the Mason–Stothers theorem is the analog of Fermat's last theorem for function fields: if ''a''(''t'')''n'' + ''b''(''t'')''n'' = ''c''(''t'')''n'' for ''a'', ''b'', ''c'' polynomials over a field of characteristic not dividing ''n'' and ''n''>2 then either at least one of ''a'', ''b'', ''c'' is 0 or they are all constant. 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「Mason–Stothers theorem」の詳細全文を読む スポンサード リンク
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