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In Galois theory, a branch of mathematics, the embedding problem is a generalization of the inverse Galois problem. Roughly speaking, it asks whether a given Galois extension can be embedded into a Galois extension in such a way that the restriction map between the corresponding Galois groups is given. ==Definition== Given a field ''K'' and a finite group ''H'', one may pose the following question (the so called inverse Galois problem). Is there a Galois extension ''F/K'' with Galois group isomorphic to ''H''. The embedding problem is a generalization of this problem: Let ''L/K'' be a Galois extension with Galois group ''G'' and let ''f'' : ''H'' → ''G'' be an epimorphism. Is there a Galois extension ''F/K'' with Galois group ''H'' and an embedding ''α'' : ''L'' → ''F'' fixing ''K'' under which the restriction map from the Galois group of ''F/K'' to the Galois group of ''L/K'' coincides with ''f''? Analogously, an embedding problem for a profinite group ''F'' consists of the following data: Two profinite groups ''H'' and ''G'' and two continuous epimorphisms ''φ'' : ''F'' → ''G'' and ''f'' : ''H'' → ''G''. The embedding problem is said to be finite if the group ''H'' is. A solution (sometimes also called weak solution) of such an embedding problem is a continuous homomorphism ''γ'' : ''F'' → ''H'' such that ''φ'' = ''f'' ''γ''. If the solution is surjective, it is called a proper solution. 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「Embedding problem」の詳細全文を読む スポンサード リンク
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