Abstract
E. Artin characterized the maximal real algebraic number field K= Q n R by its absolute Galois group Gal(Q/K) [ 1). J. Neukirch proved p-adic analogues of Artin’s theorem, and using those, he showed that a finite normal algebraic number field k is uniquely determined by its absolute Galois group Gal(Q/k), or more precisely, by the Galois group Ga@/k) of its solvable closure k [4, 51. Using Neukirch’s results, K. Uchida proved that a finite algebraic number field k is uniquely determined by Gal@k) up to isomorphisms [7]. In this paper, using the notion of “p-closed extension,” we generalize Neukirch’s p-adic analogues of Artin’s theorem and give a refinement of Uchida’s theorem.
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