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A Note on Certain Properties of Algebraic Number Fields
https://osaka-kyoiku.repo.nii.ac.jp/records/2069736
https://osaka-kyoiku.repo.nii.ac.jp/records/206973632ffb6db-7019-483c-954d-89fd81b01477
| Item type | 紀要論文 / Departmental Bulletin Paper(1) | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 公開日 | 2009-12-07 | |||||||||
| タイトル | ||||||||||
| タイトル | A Note on Certain Properties of Algebraic Number Fields | |||||||||
| 言語 | en | |||||||||
| 著者 |
NAKATSUCHI, Shūgo
× NAKATSUCHI, Shūgo
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| ISSN | ||||||||||
| 収録物識別子タイプ | ISSN | |||||||||
| 収録物識別子 | 03737411 | |||||||||
| 書誌レコードID | ||||||||||
| 収録物識別子タイプ | NCID | |||||||||
| 収録物識別子 | AN00028200 | |||||||||
| 書誌情報 |
ja : 大阪教育大学紀要 第III部門 自然科学 en : Memoirs of Osaka Kyoiku University III Natural Science and Applied Science 巻 17, 号 1, p. 1-10, 発行日 1969-01-30 |
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| 出版者 | ||||||||||
| 出版者 | 大阪教育大学 | |||||||||
| 言語 | ja | |||||||||
| 内容記述 | ||||||||||
| 内容記述タイプ | Other | |||||||||
| 内容記述 | It should be one of the most interesting themes of algebraic number theory to make clear the mutual determination (i.e. reciprocity) between certain sets of prime numbers and algebraic objects connected with an algebraic number field. Kronecker sketched a general view of the basic matters concerning this theme. And then, by Frobenius, Hurwitz, Tschebotareff, Bauer, Gaßmann, Hasse, Schmidt, and Deuring, the points of problem sketched by Kronecker were recomposed in detail and generalized. In this note I want to report some of the results which I perceived in studying their works. Notations used here are as follows: k=(an algebraic number field, i.e. a finite extension of the rational number field). Ω/k: an arb^i trary finite extension of k. m=〔Ω:k〕=(the relative degree of Ω/k). K/k: the minimal relative Galois extension of k which contains Ω, i.e. the composition of all relative conjugate fields Ω^<(i)> (i=1, 2,……, m) of Ω/k, namely, K=Ω^<(1)>Ω^<(2)>……Ω^<(m)>. K′/k: the maximal relative Galois extension of k contained in Ω, i.e. the intersection of all conjugate fields of Ω/k, namely, K′=Ω^<(1)>_∩Ω^<(2)>_∩……_∩Ω^<(m)>. R(Ω/k)=(the set of all prime ideals in k which completely split into products of distinct prime factors of relative degree 1 in Ω). The parentheses of R(●*) shows dependency on●*. P(Ω/k)=(the set of all prime ideals in k which have at least one prime factor of relative degree 1 in Ω/k, i.e. which can be descrided as relative norms of prime ideals in Ω/k). N/k : an arbitrary relative Galois extension containing Ω.〓=(the Galois group of N/k). Throughout this note, when R(Ω/k) or P(Ω/k) appears īn an assertion, it will be required, unless otherwise provided, to omit a set of exceptional prime ideals of Dirichlet-Kronecker's density zero, specially prime ideals of absolute degree > 1 and prime factors of all discriminants of fields (of a finite number) related with the assertions. Bauer's theorem(I)(see §2.)shows that the K/k, i.e. the minimal Galois extension containing Ω,is uniquely determined by R(Ω/k), and at the same time that the set R(K/k)(=R(Ω/k)) can unify all Ω/k in such a way that K/k is the minimal Galois extension containing Ω. As for P(Ω/k), a Gaßmann's result implies that the set P(Ω/k) is not always valid in determining Ω/k up to conjugates. It follows, however, according to Bauer's theorem(I●*)(see §2.), that the K′/k, i.e. the maximal Galois extension contained in Ω, is uniquely determined by P(Ω/k). It seems to me, then, that it is a point of question whether P(Ω/k) can determine the K/k uniquely in general cases. But, so far as I know, it has not been mentioned expressly in any paper till now and I want in this note to show by means of a simple disproof that P(Ω/k) can not determine K/k uniquely.●* Still it remains an open question whether the number of distinct extensions Ω/k corresponding to one and the same regular domain(i.e. P(Ω/k)) is always finite. I hope to report on this qnestion another day. In §3, I want to report a simplification of the original proof of Gaßmann's theorem in relation to a fact that the sort of split of 〓 in Ω/k can be ruled by the compartment (i.e. “Abteilung“) of Galois group 〓 of N/k. | |||||||||
| 言語 | ||||||||||
| 言語 | eng | |||||||||
| 資源タイプ | ||||||||||
| 資源タイプ | Article | |||||||||
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| Text | ||||||||||
| 資料種別(NIIタイプ) | ||||||||||
| 資料種別(NIIタイプ)識別子 | http://purl.org/coar/resource_type/c_6501 | |||||||||
| 資料種別(NIIタイプ) | departmental bulletin paper | |||||||||
| TD番号 | ||||||||||
| TD00010252 | ||||||||||
