0317１３２人目の素数さん2020/04/24(金) 22:12:29.68ID:zkcRGLXo ＞＞＞＞＞＞＞＞IUT理論7論文中に rad などという語は一つもないようだが、 WikipediaのABC予想では rad という表現が用いられている。 ただしradicallyという語が1か所だけ使われている
>Dale says: >April 23, 2020 at 11:34 pm >Kirti Joshi has now posted a revised manuscript ”On Mochizuki’s idea of Anabelomorphy and its applications” discussed earlier in this thread. >https://arxiv.org/abs/2003.01890
http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf INTER-UNIVERSAL TEICHMULLER THEORY IV:LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki April 2020 (抜粋) P1 Abstract. The present paper forms the fourth and final paper in a series of papers concerning “inter-universal Teichm¨uller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logtheta-lattice, a highly non-commutative two-dimensional diagram of “miniature models of conventional scheme theory”, called Θ±ellNF-Hodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θ-data.
P67 Section 3: Inter-universal Formalism: the Language of Species
１．上記で ”P67 Section 3: Inter-universal Formalism: the Language of Species” が 「組合せ論的カスプ化（前回04月09日の報告を参照）の論文が完成した 星裕一郎氏との共同研究でこの単射性を証明することが できそうになった。この共同研究が完成すると、松本氏の定理のproper な場合への拡張ができたことになる。 スキーム論の枠組に留まる限りとてもできそうな感じがしなかったproper な場合が、スキーム論に「パターンのヒント」を得ながらスキーム論の 枠組の外にある組合せ論的な理論を適用することによってすんなり解決 できたこと。」 とありまして
組合せ論的 スキーム論に「パターンのヒント」を得ながら って話が、Formalism: the Language of Species かなと思うわけです 0362現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/25(土) 12:11:12.82ID:O66M8Xgn>>361 思うに、ヤジウマとしては IUT IVの ”P67 Section 3: Inter-universal Formalism: the Language of Species” は、ちょっと面白なと思うわけ つまり、望月IUTの山 9000ｍ級を、IからIVと登ってきて ようやく山頂に来て、ほっと一息 山頂から降り返ってみると ”Section 3: Inter-universal Formalism: the Language of Species”をちょっと書いてみようと思ったんだろうね
で、その後、IVのAbstractが、全体のまとめになっている気がする なので、まずIVのAbstractと”P67 Section 3: Inter-universal Formalism: the Language of Species”を読むのが お薦めと思う
さて （>>166より） >ΘとかΘ±ell とか D-Θ±ell ってどう翻訳すればいいんだろうな？ とか聞かれて >>170で回答したけど IUTには、Θ±ellの説明がない 結局、山下サーベイ論文をみて分かったのだが、人に分からせようという書き方じゃないと思った（＾＾； （http://www.kurims.kyoto-u.ac.jp/~gokun/DOCUMENTS/abc2019Jul5.pdf A PROOF OF THE ABC CONJECTURE AFTER MOCHIZUKI By Go Yamashita preprint. last updated on 8/July/2019.）
IUT IV Section 3 お薦めと言った手前ご注意 グロタンディーク宇宙とか集合論拘りすぎと思う スルーした方が良い
<グロタンディーク宇宙 at IUT IV> http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf INTER-UNIVERSAL TEICHMULLER THEORY IV:LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki April 2020 (抜粋) P68 On the other hand, by the axiom of foundation, there do not exist infinite descending chains of universes V0 ∋ V1 ∋ V2 ∋ V3 ∋ ... ∋ Vn ∋ ... ? where n ranges over the natural numbers.
Bibliography [McLn] S. MacLane, One Universe as a Foundation for Category Theory, Reports of the Midwest Category Seminar III, Lecture Notes in Mathematics 106, SpringerVerlag (1969).
https://ja.wikipedia.org/wiki/%E5%9C%8F_(%E6%95%B0%E5%AD%A6) 圏 (数学) (抜粋) 圏の大きさ 圏 C が小さい (small) とは、対象の類 ob(C) および射の類 hom(C) がともに集合となる（つまり真の類でない）ときに言い、さもなくば大きい (large) と言う。射の類が集合とならずとも、任意の二対象 a, b ∈ ob(C) をとるごとに、射の類 hom(a, b) が集合となるならば（hom(a, b) を射集合、ホム集合などと呼び）、その圏は局所的に小さい (locally small) と言う[3]。 集合の圏など数学における重要な圏の多くは、小さくないとしても、少なくとも局所的に小さい。 文献によっては、局所的に小さい圏のみを扱い、それを単に圏と呼ぶ場合もある。
>On the other hand, by the axiom of foundation, there do not exist infinite descending chains of universes >V0 ∋ V1 ∋ V2 ∋ V3 ∋ ... ∋ Vn ∋ ...
ここ、普通は、the axiom of foundationは、集合の無限降下列をいう ”there do not exist infinite descending chains of universes”と、「universes」でいう意味が薄い （余計なシッタカでしょ）
>Bibliography >[McLn] S. MacLane, One Universe as a Foundation for Category Theory, Reports of the >Midwest Category Seminar III, Lecture Notes in Mathematics 106, SpringerVerlag (1969).
古い、古すぎる 確か、「2-圏」とかIUT中にあったけど、一覧表では 「大きい 自然変換も考えると2-圏（英語版）の例となる」 とあるから、「グロタンディーク宇宙」に拘る理由がない気がする 1969年 と2020年（今）とでは、時代が違いすぎる 今とでは、圏論のレベルと普及度が違いすぎると思う 0377現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/25(土) 17:57:03.99ID:b0fzLo6k 下記ね、”This paper of Joshi”をこき下ろしているのだが これ見て、Joshi さんが、「ごらぁ〜！」と怒鳴り込んで、反論してバトルになって それにショルツ先生も参加してバトルしてくれると、面白いね、ヤジウマとしてはｗ（＾＾；
https://www.math.columbia.edu/~woit/wordpress/?p=11709&cpage=3#comment-236147 Fierce Inertia says: April 24, 2020 at 10:48 am This paper of Joshi is remarkably unconvincing to me. If I may caricature it slightly, it seems to only contain the following types of results: 1. Statements of the form “(Thing X / Property Y) depends only on the absolute Galois group of a p-adic field.” None of these are surprising or difficult: they all follow from basic class field theory or from the Jannsen-Wingberg theorem (which IS a difficult result, cf. here for a nice overview: http://www.numdam.org/article/AST_1982__94__153_0.pdf)
2. Statements of the form “(Thing X / Property Y) does not depend only on the absolute Galois group of a p-adic field.” These are even less surprising, and they also follow from Jannsen-Wingberg, or from five seconds of thought.
4. Vague suggestions that various things can be interpreted anabelomorphically. What evidence is there here that this perspective of anabelomorphy is actually useful? What can you DO with it? The answer this paper seems to suggest is: nothing.
I am happy to be convinced otherwise. (引用終り) 0378現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/25(土) 18:14:19.48ID:b0fzLo6k>>377補足 >Fierce Inertia says: >None of these are surprising or difficult: they all follow from basic class field theory or from the Jannsen-Wingberg theorem (which IS a difficult result, cf. here for a nice overview: http://www.numdam.org/article/AST_1982__94__153_0.pdf)
https://en.wikipedia.org/wiki/Neukirch%E2%80%93Uchida_theorem Neukirch?Uchida theorem (抜粋) In mathematics, the Neukirch?Uchida theorem shows that all problems about algebraic number fields can be reduced to problems about their absolute Galois groups. Jurgen Neukirch (1969) showed that two algebraic number fields with the same absolute Galois group are isomorphic, and Koji Uchida (1976) strengthened this by proving Neukirch's conjecture that automorphisms of the algebraic number field correspond to outer automorphisms of its absolute Galois group. Florian Pop (1990, 1994) extended the result to infinite fields that are finitely generated over prime fields. The Neukirch?Uchida theorem is one of the foundational results of anabelian geometry, whose main theme is to reduce properties of geometric objects to properties of their fundamental groups, provided these fundamental groups are sufficiently non-abelian.
Anabelian geometry って、明らかに (1982)より後で ”Before anabelian geometry proper began with the famous letter to Gerd Faltings and Esquisse d'un Programme, the Neukirch?Uchida theorem hinted at the program from the perspective of Galois groups, which themselves can be shown to be etale fundamental groups.” とある
https://en.wikipedia.org/wiki/Anabelian_geometry Anabelian geometry (抜粋) Anabelian geometry is a theory in number theory, which describes the way in which the algebraic fundamental group G of a certain arithmetic variety V, or some related geometric object, can help to restore V. The first traditional conjectures, originating from Alexander Grothendieck and introduced in Esquisse d'un Programme were about how topological homomorphisms between two groups of two hyperbolic curves over number fields correspond to maps between the curves. These Grothendieck conjectures were partially solved by Hiroaki Nakamura and Akio Tamagawa, while complete proofs were given by Shinichi Mochizuki. Before anabelian geometry proper began with the famous letter to Gerd Faltings and Esquisse d'un Programme, the Neukirch?Uchida theorem hinted at the program from the perspective of Galois groups, which themselves can be shown to be etale fundamental groups.
More recently, Mochizuki introduced and developed a so called mono-anabelian geometry which restores, for a certain class of hyperbolic curves over number fields, the curve from its algebraic fundamental group. Key results of mono-anabelian geometry were published in Mochizuki's "Topics in Absolute Anabelian Geometry."
Contents 1 Formulation of a conjecture of Grothendieck on curves 2 See also
See also ・Fiber functor ・Neukirch?Uchida theorem ・Belyi's theorem
Notes 1^ Schneps, Leila (1997). "Grothendieck's "Long march through Galois theory"". In Schneps; Lochak, Pierre (eds.). Geometric Galois actions. 1. London Mathematical Society Lecture Note Series. 242. Cambridge: Cambridge University Press. pp. 59?66. MR 1483109. 2^ Mochizuki, Shinichi (1996). "The profinite Grothendieck conjecture for closed hyperbolic curves over number fields". J. Math. Sci. Univ. Tokyo. 3 (3): 571?627. hdl:2261/1381. MR 1432110. 3^ Ihara, Yasutaka; Nakamura, Hiroaki (1997). "Some illustrative examples for anabelian geometry in high dimensions" (PDF). In Schneps, Leila; Lochak, Pierre (eds.). Geometric Galois actions. 1. London Mathematical Society Lecture Note Series. 242. Cambridge: Cambridge University Press. pp. 127?138. MR 1483114. 4^ Mochizuki, Shinichi (2003). "The absolute anabelian geometry of canonical curves" (PDF). Documenta Mathematica. Extra Vol., Kazuya Kato's fiftieth birthday: 609?640. MR 2046610. (引用終り) 以上 0381現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/25(土) 19:16:07.56ID:b0fzLo6k>>379
日本語版wikipedia では、”Neukirch-Uchida”への直接の言及がないな（＾＾；
https://ja.wikipedia.org/wiki/%E9%81%A0%E3%82%A2%E3%83%BC%E3%83%99%E3%83%AB%E5%B9%BE%E4%BD%95%E5%AD%A6 遠アーベル幾何学 (抜粋) 遠アーベル幾何学(Anabelian geometry)は数学の理論であり、代数多様体 V 上の代数的基本群（英語版）(algebraic fundamental group) G や関連する幾何学的対象を記述する。また、V をどのように他の幾何学的対象 W へ写像することができるかを決定する。 いずれもより詳細な意味は、G がアーベル群から非常に遠い場合を前提とするという意味である。単語としての遠アーベル（アーベルの前に、接頭語である an がついたもの）は、1980年代のアレクサンドル・グロタンディーク(Alexander Grothendieck)の有名な著作であるEsquisse d'un Programmeで導入された[1]。
曲線上のグロタンディークの予想の定式化 「遠アーベル的問題」とは次のように定式化される。 「 多様体 X の同型類についてのどのくらいの情報が、エタール基本群（英語版）(etale fundamental group)の知識には含まれているのであろうか？[2] 」 具体例は、多様体が射影的と同様にアフィン的な場合である。有限生成な体 K （その上の素体）上に定義された滑らかで既約な場合を想定し、与えられた双曲線 C に対し、つまり、種数 g の射影代数曲線内の n 個の点の補空間に対し、 2 - 2g - n < 0 とする。グロタンディークは、射有限群である C の代数的基本群 G が C 自身を決定する（つまり G の同型類が C の同型類を決定する）と予想した。
このことは望月新一により証明された[3] g = 0（射影直線）で n = 4 の場合の例が与えられ、このとき、C の同型類が K の中の削除される 4つの点の連比により決定される。 （ほとんど、連比で 4つの点の順序であるが、点を取り去ると存在しない。）[4] K が局所体の場合の結果もある[5]。
関連項目 ノイキルヒ・内田の定理（英語版）
脚注 1^ Alexander Grothendieck, 1984. "Esquisse d'un Programme", (1984 manuscript), http://people.math.jussieu.fr/~leila/grothendieckcircle/EsquisseFr.pdf published in "Geometric Galois Actions", L. Schneps, P. Lochak, eds., London Math. Soc. Lecture Notes 242, Cambridge University Press, 1997, pp. 5-48; English transl., ibid., pp. 243-283. 2^ https://webusers.imj-prg.fr/~leila.schneps/SchnepsLM.pdf Grothendieck’s “Long March through Galois theory”Leila Schneps * The result of this transcription ? the possibility of which was referred to by Grothendieck in the Esquisse as “une compilation de notes pieusement accumul´ees” 3^ S. Mochizuki, The profinite Grothendieck conjecture for hyperbolic curves over number fields, J. Math. Sci. Univ. Tokyo 3 (1996), 571?627. 5^ http://www.math.uiuc.edu/documenta/vol-kato/mochizuki.dm.pdf (引用終り) 以上 0383現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/25(土) 19:24:47.12ID:b0fzLo6k>>382 ＞遠アーベル幾何学
S. Mochizuki,の存在が大きいね それと、 Ihara, Yasutaka先生ね
(上記へ追加) http://www.math.sci.osaka-u.ac.jp/~nakamura/zoo/lion/INanabel.pdf 3^ Ihara, Yasutaka; Nakamura, Hiroaki (1997). "Some illustrative examples for anabelian geometry in high dimensions" (PDF). In Schneps, Leila; Lochak, Pierre (eds.). Geometric Galois actions. 1. London Mathematical Society Lecture Note Series. 242. Cambridge: Cambridge University Press. pp. 127?138. MR 1483114. 0384現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/25(土) 19:35:44.83ID:b0fzLo6k>>378 補足 (引用開始) 引用のPDFは、JURGEN NEUKIRCHとあって、有名な”ノイキルヒ 内田”の人でしょ？（下記） で、 (1982)ってのがね〜ｗ Fierce Inertia のいうことにゃ、Joshi の書いてあることは、 (1982)と同じだと それって、ショルツ先生が、IUTに対して行った誤読に似てるんじゃない？ つまり、勝手に単純化した解釈して、 (1982)JURGEN NEUKIRCHで終りって 「怒れ！ Joshi！」って煽ったりして (引用終り)
https://arxiv.org/abs/2003.01890 Dale says: April 23, 2020 at 11:34 pm Kirti Joshi has now posted a revised manuscript ”On Mochizuki’s idea of Anabelomorphy and its applications” discussed earlier in this thread. 0385現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/25(土) 21:14:24.65ID:b0fzLo6k>>384 補足の補足
”https://arxiv.org/abs/2003.01890 Dale says: April 23, 2020 at 11:34 pm Kirti Joshi has now posted a revised manuscript ”On Mochizuki’s idea of Anabelomorphy and its applications” discussed earlier in this thread.”
今一度、ざっと目を通したけど 意味分からんけどｗ（＾＾
”Fierce Inertia ”の主張は当たってない気がする まあ、”Kirti Joshi ”の怒鳴り込みを待ちましょうｗ（＾＾； 0386現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/25(土) 23:19:12.26ID:b0fzLo6k>>379 >Before anabelian geometry proper began with the famous letter to Gerd Faltings
・Grothendiecks Brief an Faltings uber Anabelsche Geometrie von 1983 findet sich hier: Online http://www.grothendieckcircle.org/ http://www.math.jussieu.fr/~leila/grothendieckcircle/mathtexts.php Mathematical Texts Correspondence -- A selection of letters from Grothendieck to various people
Anabelian letter to Faltings (June 27, 1983) - A letter from Grothendieck to G. Faltings in German, describing Anabelian Algebraic Geometry. To Grothendieck's disappointment, Faltings never responded to this letter. However, Faltings' student Shinichi Mochizuki picked up the subject years later and proved Grothendieck's anabelian conjecture for hyperbolic curves. This letter was later published in Geometric Galois Actions I (P. Lochak and L. Schneps, eds., London Math Society Lecture Note Series 242, Cambridge University Press (2000) ). (English translation) http://www.math.jussieu.fr/~leila/grothendieckcircle/Letters/GtoF.pdf ( Scan of the original) http://www.math.jussieu.fr/~leila/grothendieckcircle/Letters/falt.html 0387現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/25(土) 23:38:25.42ID:b0fzLo6k ”多項式の解の近似がとりもつ数論と幾何の関係 (1), (2), (3), (4).” これ、調べると数学セミナー 2000.4〜2000.7 の文章と分かった
（参考：Ivan Fesenko氏のコメントPDF書誌） https://www.maths.nottingham.ac.uk/plp/pmzibf/ Ivan Fesenko https://www.maths.nottingham.ac.uk/plp/pmzibf/nov.html News - Ivan Fesenko ・On pioneering mathematical research, on the occasion of the announced publication of the IUT papers by Shinichi Mochizuki, April 2020 media1 media2 media3 media4 https://www.maths.nottingham.ac.uk/plp/pmzibf/rpp.pdf ON PIONEERING MATHEMATICAL RESEARCH, ON THE OCCASION OF ANNOUNCEMENT OF FORTHCOMING PUBLICATION OF THE IUT PAPERS BY SHINICHI MOCHIZUKI IVAN FESENKO Date: April 3 2020 0391１３２人目の素数さん2020/04/26(日) 06:10:14.48ID:inFd57k6>>370 デｨｯヂテェｪ−ｯ!って書きそうw 0392現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/26(日) 06:23:11.19ID:7O7a3CML>>390 補足
いま見ると、Woitブログの冒頭にあるね（＾＾；
https://www.math.columbia.edu/~woit/wordpress/?p=11709 Not Even Wrong Latest on abc Posted on April 3, 2020 by woit (抜粋) Ivan Fesenko today has a long article entitled https://www.maths.nottingham.ac.uk/plp/pmzibf/rpp.pdf On Pioneering Mathematical Research, On the Occasion of Announcement of Forthcoming Publication of the IUT Papers by Shinichi Mochizuki. Much like earlier articles from him (I’d missed this one), it’s full of denunciations of anyone (including Scholze) who has expressed skepticism about the proof as an incompetent. There’s a lot about how Mochizuki’s work on the purported proof is an inspiration to the world, ending with:
In the UK, the recent new additional funding of mathematics, work on which was inspired by the pioneering research of Sh. Mochizuki, will address some of these issues.
which refers to the British government decision discussed here. 0393現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/26(日) 06:39:34.68ID:7O7a3CML>>391 どうも、コメントありがとう（＾＾ 0394現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/26(日) 07:57:32.82ID:7O7a3CML>>387 追加
https://books.google.co.jp/books?id=yJwNrABugDEC&;printsec=frontcover&dq=Matsumura,+Commutative+Algebra&hl=ja&sa=X&ved=0ahUKEwiJv6GP14TpAhUVfnAKHZbpDvoQ6AEIKzAA#v=onepage&q=Matsumura%2C%20Commutative%20Algebra&f=false Commutative Ring Theory H. Matsumura, ?B. Bollobas - 1989 - ?プレビュー 10件あり
http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf Remark 3.1.4. Note that because the data involved in a species is given by abstract set-theoretic formulas, the mathematical notion constituted by the species is immune to, i.e., unaffected by, extensions of the universe - i.e., such as the ascending chain V0 ∈ V1 ∈ V2 ∈ V3 ∈ ... ∈ Vn ∈ ... ∈ V that appears in the discussion preceding Definition 3.1 - in which one works. This is the sense in which we apply the term “inter-universal”. That is to say, “inter-universal geometry” allows one to relate the “geometries” that occur in distinct universes.
https://ncatlab.org/nlab/show/David+Michael+Roberts nLab David Michael Roberts (抜粋) 1. Writing ・The formal construction of formal anafunctors (2018), arXiv:1808.04552 doi:10.25909/5b6cfd1a73e55 (Note that this was cited in Internal Categories, Anafunctors and Localisations with the title Strict 2-sites, J-spans and Localisations, and some paper containing these notes may yet have that title) Submitted. ・Smooth loop stacks of differentiable stacks and gerbes, Cahiers de Topologie et Geometrie Differentielle Categoriques, Vol LIX no 2 (2018) pp 95-141 journal version, arXiv:1602.07973. Joint with Raymond Vozzo. ・On certain 2-categories admitting localisation by bicategories of fractions, Applied Categorical Structures Volume 24, Issue 4 (2016) pp 373-384, doi:10.1007/s10485-015-9400-4, ReadCube, arXiv:1402.7108. ・The weak choice principle WISC may fail in the category of sets, Studia Logica Volume 103, Issue 5 (2015) pp 1005-1017, doi:10.1007/s11225-015-9603-6 arXiv:1311.3074. ・Internal categories, anafunctors and localisations, Theory and Applications of Categories, Vol. 26, 2012, No. 29, pp 788-829, journal version, arXiv:1101.2363
0! = 1 1! = 1 2! = 2 x 1 = 2 3! = 3 x 2 x 1 = 6 4! = 4 x 3 x 2 x 1 = 24 … 10! = 10 x 9 x … x 2 x 1 = 3,628,800 0413現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/26(日) 12:22:34.21ID:7O7a3CML>>406 >ノイキルヒ本とは、分岐点がノイキルヒが2点で
分岐 ”Ramification theory of valuations” （参考） https://en.wikipedia.org/wiki/Ramification_theory_of_valuations Ramification theory of valuations (抜粋) In mathematics, the ramification theory of valuations studies the set of extensions of a valuation v of a field K to an extension L of K. It is a generalization of the ramification theory of Dedekind domains.
Contents 1 Galois case 1.1 Decomposition group and inertia group 2 See also 3 References Galois case The structure of the set of extensions is known better when L/K is Galois.
Decomposition group and inertia group Let (K, v) be a valued field and let L be a finite Galois extension of K. Let Sv be the set of equivalence classes of extensions of v to L and let G be the Galois group of L over K. Then G acts on Sv by σ[w] = [w ? σ] (i.e. w is a representative of the equivalence class [w] ∈ Sv and [w] is sent to the equivalence class of the composition of w with the automorphism σ : L → L; this is independent of the choice of w in [w]). In fact, this action is transitive.
Given a fixed extension w of v to L, the decomposition group of w is the stabilizer subgroup Gw of [w], i.e. it is the subgroup of G consisting of all elements that fix the equivalence class [w] ∈ Sv.
Let mw denote the maximal ideal of w inside the valuation ring Rw of w. The inertia group of w is the subgroup Iw of Gw consisting of elements σ such that σx ≡ x (mod mw) for all x in Rw. In other words, Iw consists of the elements of the decomposition group that act trivially on the residue field of w. It is a normal subgroup of Gw.
The reduced ramification index e(w/v) is independent of w and is denoted e(v). Similarly, the relative degree f(w/v) is also independent of w and is denoted f(v).
See also ramification group 0414現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/04/26(日) 12:27:59.81ID:7O7a3CML>>413 >See also >ramification group
(参考) https://en.wikipedia.org/wiki/Ramification_group Ramification group (抜粋) In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension. Contents 1 Ramification groups in lower numbering 1.1 Example: the cyclotomic extension 1.2 Example: a quartic extension 2 Ramification groups in upper numbering 2.1 Herbrand's theorem 3 See also 4 Notes 5 References
Example: the cyclotomic extension The ramification groups for a cyclotomic extension {\displaystyle K_{n}:=\mathbf {Q} _{p}(\zeta )/\mathbf {Q} _{p}}{\displaystyle K_{n}:=\mathbf {Q} _{p}(\zeta )/\mathbf {Q} _{p}}, where {\displaystyle \zeta }\zeta is a {\displaystyle p^{n}}p^{n}-th primitive root of unity, can be described explicitly:[9]
{\displaystyle G_{s}=Gal(K_{n}/K_{e}),}{\displaystyle G_{s}=Gal(K_{n}/K_{e}),} where e is chosen such that {\displaystyle p^{e-1}\leq s<p^{e}}{\displaystyle p^{e-1}\leq s<p^{e}}.
Example: a quartic extension Let K be the extension of Q2 generated by {\displaystyle x_{1}={\sqrt {2+{\sqrt {2}}\ }}}{\displaystyle x_{1}={\sqrt {2+{\sqrt {2}}\ }}}. The conjugates of x1 are x2={\displaystyle x_{2}={\sqrt {2-{\sqrt {2}}\ }},}{\displaystyle x_{2}={\sqrt {2-{\sqrt {2}}\ }},} x3 = ?x1, x4 = ?x2.
