ON INFINITE CHAINS OF INTERMEDIATE FIELDS

Authors : David E Dobbs

Pages : 165-176

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Publication Date : 2012-06-01

Article Type : Research Paper

Abstract :If K ⊂ L is an infinite-dimensional algebraic field extension and the insert ignore into journalissuearticles values(infinite); cardinal number ℵ := [L : K] insert ignore into journalissuearticles values(the K-vector space dimension of L);, then there exists an infinite maximal chain, C, consisting of fields contained between K and L, such that the cardinality of C is at most ℵ. If K ⊂ L is a J-extension, then every maximal chain of intermediate fields has cardinality ℵ0. However, an example is given where K ⊂ L has maximal chains, D and E, of intermediate fields such that the cardinalities of D and E are ℵ and 2ℵ, respectively.
Keywords : field extension, intermediate field, maximal chain, infinite cardinal number, algebraic extension, transfinite induction, J extension, transcendence degree, integral domain, classical D M construction