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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 Splitting of prime ideals in Galois extensions ： ウィキペディア英語版 Splitting of prime ideals in Galois extensions In mathematics, the interplay between the Galois group ''G'' of a Galois extension ''L'' of a number field ''K'', and the way the prime ideals ''P'' of the ring of integers ''O''''K'' factorise as products of prime ideals of ''O''''L'', provides one of the richest parts of algebraic number theory. The splitting of prime ideals in Galois extensions is sometimes attributed to David Hilbert by calling it Hilbert theory. There is a geometric analogue, for ramified coverings of Riemann surfaces, which is simpler in that only one kind of subgroup of ''G'' need be considered, rather than two. This was certainly familiar before Hilbert. == Definitions == Let ''L''/''K'' be a finite extension of number fields, and let ''OK'' and ''OL'' be the corresponding ring of integers of ''K'' and ''L'', respectively, which are defined to be the integral closure of the integers Z in the field in question. : Finally, let ''p'' be a non-zero prime ideal in ''OK'', or equivalently, a maximal ideal, so that the residue ''OK''/''p'' is a field. From the basic theory of one-dimensional rings follows the existence of a unique decomposition : $pO\_L\; =\; \backslash prod\_^\; P\_j^$ of the ideal ''pOL'' generated in ''OL'' by ''p'' into a product of distinct maximal ideals ''P''''j'', with multiplicities ''e''''j''. The field ''F'' = ''OK''/''p'' naturally embeds into ''F''''j'' = ''OL''/''P''''j'' for every ''j'', the degree ''f''''j'' = (: ''OK''/''p'' ) of this residue field extension is called inertia degree of ''P''''j'' over ''p''. The multiplicity ''e''''j'' is called ramification index of ''P''''j'' over ''p''. If it's bigger than 1 for some ''j'', the field extension ''L''/''K'' is called ramified at ''p'' (or we say that ''p'' ramifies in ''L'', or that it is ramified in ''L''). Otherwise, ''L''/''K'' is called unramified at ''p''. If this is the case then by the Chinese remainder theorem the quotient ''OL''/''pOL'' is a product of fields ''F''''j''. The extension ''L''/''K'' is ramified in exactly those primes that divide the relative discriminant, hence the extension is unramified in all but finitely many prime ideals. Multiplicativity of ideal norm implies : $()=\backslash sum\_^\; e\_j\; f\_j.$ If ''f''''j'' = ''e''''j'' = 1 for every ''j'' (and thus ''g'' = (: ''K'' )), we say that ''p'' splits completely in ''L''. If ''g'' = 1 and ''f''''1'' = 1 (and so ''e''''1'' = (: ''K'' )), we say that ''p'' ramifies completely in ''L''. Finally, if ''g'' = 1 and ''e''''1'' = 1 (and so ''f''''1'' = (: ''K'' )), we say that ''p'' is inert in ''L''. 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「Splitting of prime ideals in Galois extensions」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.