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In mathematics, in the field of algebraic number theory, a Bauerian extension is a field extension of an algebraic number field which is characterized by the prime ideals with inertial degree one in the extension. For a finite degree extension ''L''/''K'' of an algebraic number field ''K'' we define ''P''(''L''/''K'') to be the set of primes p of ''K'' which have a factor P with inertial degree one (that is, the residue field of P has the same order as the residue field of p). Bauer's theorem states that if ''M''/''K'' is a finite degree Galois extension, then ''P''(''M''/''K'') ⊇ ''P''(''L''/''K'') if and only if ''M'' ⊆ ''L''. In particular, finite degree Galois extensions ''N'' of ''K'' are characterised by set of prime ideals which split completely in ''N''. An extension ''F''/''K'' is ''Bauerian'' if it obeys Bauer's theorem: that is, for every finite extension ''L'' of ''K'', we have ''P''(''F''/''K'') ⊇ ''P''(''L''/''K'') if and only if ''L'' contains a subfield ''K''-isomorphic to ''F''. All field extensions of degree at most 4 over Q are Bauerian.〔Narkiewicz (1990) p.416〕 An example of a non-Bauerian extension is the Galois extension of Q by the roots of 2''x''5 − 32''x'' + 1, which has Galois group ''S''5.〔Narkiewicz (1990) p.394〕 ==See also== * Splitting of prime ideals in Galois extensions 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「Bauerian extension」の詳細全文を読む スポンサード リンク
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