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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Minimal polynomial (field theory) ： ウィキペディア英語版 Minimal polynomial (field theory) In field theory, a branch of mathematics, a minimal polynomial is defined relative to a field extension ''E/F'' and an element of the extension field ''E''. The minimal polynomial of an element, if it exists, is a member of ''F''(), the ring of polynomials in the variable ''x'' with coefficients in ''F''. Given an element ''α'' of ''E'', let ''J''''α'' be the set of all polynomials ''f''(''x'') in ''F''() such that ''f''(''α'') = 0. The element ''α'' is called a root or zero of each polynomial in ''J''''α''. The set ''J''''α'' is so named because it is an ideal of ''F''(). The zero polynomial, whose every coefficient is 0, is in every ''J''''α'' since 0''α''''i'' = 0 for all ''α'' and ''i''. This makes the zero polynomial useless for classifying different values of ''α'' into types, so it is excepted. If there are any non-zero polynomials in ''J''''α'', then ''α'' is called an algebraic element over ''F'', and there exists a monic polynomial of least degree in ''J''''α''. This is the minimal polynomial of ''α'' with respect to ''E''/''F''. It is unique and irreducible over ''F''. If the zero polynomial is the only member of ''J''''α'', then ''α'' is called a transcendental element over ''F'' and has no minimal polynomial with respect to ''E''/''F''. Minimal polynomials are useful for constructing and analyzing field extensions. When ''α'' is algebraic with minimal polynomial ''a''(''x''), the smallest field that contains both ''F'' and ''α'' is isomorphic to the quotient ring ''F''()/⟨''a''(''x'')⟩, where ⟨''a''(''x'')⟩ is the ideal of ''F''() generated by ''a''(''x''). Minimal polynomials are also used to define conjugate elements. == Definition == Let ''E''/''F'' be a field extension, α an element of ''E'', and ''F''() the ring of polynomials in ''x'' over ''F''. The minimal polynomial of ''α'' is the monic polynomial of least degree among all polynomials in ''F''() having ''α'' as a root; it exists when ''α'' is algebraic over ''F'', that is, when ''f''(''α'') = 0 for some non-zero polynomial ''f''(''x'') in ''F''(). 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「Minimal polynomial (field theory)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.