[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む78 (1002ﾚｽ)
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366(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)07:01 ID:ZLEqKHqI(1/18) AAS
>>357
> The English term "field" was introduced by Moore (1893).[19]
E. H. Moore（1862 ? 1932)さん
("Robert Lee Moore (no relation) Topologist (1882 ? 1974) " は、別人ですね)
（参考）
外部ﾘﾝｸ:en.wikipedia.org
(抜粋)
省10
367: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)07:02 ID:ZLEqKHqI(2/18) AAS
>>366
つづき
Topologist
According to the bibliography in Wilder (1976), Moore published 67 papers and one monograph, his 1932 Foundations of Point Set Theory.
He is primarily remembered for his work on the foundations of topology, a topic he first touched on in his Ph.D. thesis.
By the time Moore returned to the University of Texas, he had published 17 papers on point-set topology?a term he coined?including his 1915 paper "On a set of postulates which suffice to define a number-plane", giving an axiom system for plane topology.
The Moore plane, Moore's road space, Moore space, Moore's quotient theorem [ru] and the Moore space conjecture are named in his honor.
省2
369: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)07:54 ID:ZLEqKHqI(3/18) AAS
>>353 補足
モジュラー群の歴史で
和文：しかし、密接に関連する楕円曲線は、1785年にジョゼフ＝ルイ・ラグランジュ(Joseph Louis Lagrange)により研究され、
さらに楕円函数に関する結果は、カール・グスタフ・ヤコブ・ヤコビ(Carl Gustav Jakob Jacobi)とニールス・アーベル(Niels Henrik Abel)により1827年に出版された。
↑
英文：However, the closely related elliptic functions were studied by Joseph Louis Lagrange in 1785,
and further results on elliptic functions were published by Carl Gustav Jakob Jacobi and Niels Henrik Abel in 1827.
省17
370: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)07:55 ID:ZLEqKHqI(4/18) AAS
>>368
あんたはそれ以下
371(2): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)08:00 ID:ZLEqKHqI(5/18) AAS
なにを言っているか分からない
自分が、数学の専門誌（レフェリー）に、論文を投稿したのか？
なら、ハナタカも結構だけどね
”おまえはコピペだけだが、おれは数学を深く理解している”とでも言いたいのか？
笑えるよ
自慢する場所を間違えているぜ
５ｃｈの数学板で、「おれは分かっている」とハナタカかい？ｗ（＾＾
省4
375(8): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)10:29 ID:ZLEqKHqI(6/18) AAS
>>325
(引用開始)
「qを有限体の位数として
PSL(2,q)はP1(Fq)への推移的な作用で
P1(Fq)の位数はq+1だから
PSL(2,q)はq+1次対称群に埋め込める」
が正しい
省34
376(3): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)10:30 ID:ZLEqKHqI(7/18) AAS
>>375
つづき
Action on projective line
Some of the above maps can be seen directly in terms of the action of PSL and PGL on the associated projective line:
PGL(n, q) acts on the projective space Pn?1(q), which has (q^n?1)/(q?1) points,
and this yields a map from the projective linear group to the symmetric group on (q^n?1)/(q?1) points.
For n = 2, this is the projective line P1(q) which has (q^2?1)/(q?1) = q+1 points,
省13
377(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)10:33 ID:ZLEqKHqI(8/18) AAS
>>376
つづき
・PSL(2, 5) < PGL(2, 5) → S6, of orders 60 and 120, which yields an embedding of S5 (respectively, A5) as a transitive subgroup of S6 (respectively, A6). This is an example of an exotic map S5 → S6,
and can be used to construct the exceptional outer automorphism of S6.[6] Note that the isomorphism PGL(2, 5) =~ S5 is not transparent from this presentation: there is no particularly natural set of 5 elements on which PGL(2, 5) acts.
Action on p points
While PSL(n, q) naturally acts on (qn?1)/(q?1) = 1+q+...+qn?1 points, non-trivial actions on fewer points are rarer. I
ndeed, for PSL(2, p) acts non-trivially on p points if and only if p = 2, 3, 5, 7, or 11; for 2 and 3 the group is not simple, while for 5, 7, and 11, the group is simple - further, it does not act non-trivially on fewer than p points.[note 5] This was first observed by Evariste Galois in his last letter to Chevalier, 1832.[7]
省8
378(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)10:34 ID:ZLEqKHqI(9/18) AAS
>>377
つづき
One then needs to consider the action of the symmetry group of icosahedron on the five associated tetrahedra.
・L2(7) =~ L3(2) which acts on the 1+2+4 = 7 points of the Fano plane (projective plane over F2); this can also be seen as the action on order 2 biplane, which is the complementary Fano plane.
・L2(11) is subtler, and elaborated below; it acts on the order 3 biplane.[8]
Further, L2(7) and L2(11) have two inequivalent actions on p points; geometrically this is realized by the action on a biplane,
which has p points and p blocks - the action on the points and the action on the blocks are both actions on p points,
省7
379(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)10:35 ID:ZLEqKHqI(10/18) AAS
>>378
つづき
The action of L2(11) can be seen algebraically as due to an exceptional inclusion L2(5)→ L2(11) - there are two conjugacy classes of subgroups of L2(11) that are isomorphic to L2(5),
each with 11 elements: the action of L2(11) by conjugation on these is an action on 11 points,
and, further, the two conjugacy classes are related by an outer automorphism of L2(11). (The same is true for subgroups of L2(7) isomorphic to S4, and this also has a biplane geometry.)
Geometrically, this action can be understood via a biplane geometry, which is defined as follows.
A biplane geometry is a symmetric design (a set of points and an equal number of "lines", or rather blocks) such that any set of two points is contained in two lines, while any two lines intersect in two points;
省8
380(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)10:36 ID:ZLEqKHqI(11/18) AAS
>>379
つづき
More surprisingly, the coset space L2(11)/Z/11Z, which has order 660/11 = 60 (and on which the icosahedral group acts) naturally has the structure of a buckeyball, which is used in the construction of the buckyball surface.
History
The groups PSL(2, p) were constructed by Evariste Galois in the 1830s,
and were the second family of finite simple groups, after the alternating groups.[3]
Galois constructed them as fractional linear transforms, and observed that they were simple except if p was 2 or 3;
省4
381(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)10:36 ID:ZLEqKHqI(12/18) AAS
>>380
つづき
C60：1996 Nobel Prize in Chemistry でしたね
外部ﾘﾝｸ:en.wikipedia.org
Buckminsterfullerene
(Redirected from Buckeyball)
History
省25
382: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)10:37 ID:ZLEqKHqI(13/18) AAS
>>381
つづき
外部ﾘﾝｸ:en.wikipedia.org
ADE classification
(抜粋)
画像ﾘﾝｸ[png]:upload.wikimedia.org
The simply laced Dynkin diagrams classify diverse mathematical objects.
省1
383: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)10:56 ID:ZLEqKHqI(14/18) AAS
>>375 補足
まあ、当方の知識と理解に穴があるというのは、その通りだろう。認めるよ
但し、数学科生といえども、入学から卒業まで、試験は全部満点という人も、いないだろう
全部満点でなくとも、卒業できるし
穴なら埋めれば良い
ところで、
（>>324より）
省23
384(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)11:26 ID:ZLEqKHqI(15/18) AAS
>>375
> PSL(2,q)はP1(Fq)への推移的な作用で
外部ﾘﾝｸ:en.wikipedia.org
Projective space
(抜粋)
Morphisms
Injective linear maps T ∈ L(V, W) between two vector spaces V and W over the same field k induce mappings of the corresponding projective spaces P(V) → P(W) via:
省9
385(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)11:27 ID:ZLEqKHqI(16/18) AAS
>>384
つづき
Finite projective spaces and planes
Further information on finite projective planes: Projective plane § Finite projective planes
For finite projective spaces of dimension at least three, Wedderburn's theorem implies that the division ring over which the projective space is defined must be a finite field, GF(q), whose order
(that is, number of elements) is q (a prime power). A finite projective space defined over such a finite field has q + 1 points on a line, so the two concepts of order coincide. Notationally, PG(n, GF(q)) is usually written as PG(n, q).
All finite fields of the same order are isomorphic, so, up to isomorphism, there is only one finite projective space for each dimension greater than or equal to three, over a given finite field.
省5
386: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)11:28 ID:ZLEqKHqI(17/18) AAS
>>385
つづき
Scheme theory
Scheme theory, introduced by Alexander Grothendieck during the second half of 20th century, allows defining a generalization of algebraic varieties, called schemes,
by gluing together smaller pieces called affine schemes, similarly as manifolds can be built by gluing together open sets of {\displaystyle \mathbb {R} ^{n}.}{\displaystyle \mathbb {R} ^{n}.}
The Proj construction is the construction of the scheme of a projective space, and, more generally of any projective variety, by gluing together affine schemes. In the case of projective spaces, one can take for these affine schemes the affine schemes associated to the charts (affine spaces) of the above description of a projective space as a manifold.
See also: Algebraic geometry of projective spaces
省2
387(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/02(土)11:48 ID:ZLEqKHqI(18/18) AAS
>>375 追加
＞ PSL(2,q)はP1(Fq)への推移的な作用で
これ大丈夫か？
”PGL(2,q)はP1(Fq)への推移的な作用”ならば、言えると思うが
下記の”ポワンカレの上半平面モデル”より、
「この群の部分群で上半平面 H を H 自身の上に移すものは、すべての係数が実数であるような変換全体の成す群 PSL(2, R) 」
「その作用は上半平面上推移的かつ等距」とあるけどね？（＾＾；
省21
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