[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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463(2): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/04(月)17:47 ID:Qu1TcOyQ(4/28) AAS
>>462
つづき
1. Introduction
The non-abelian finite simple groups and their automorphism groups play a crucial role
in an inductive approach to the inverse problem of Galois theory. The rigidity method
(see for example Malle and Matzat (1999)) has proved very efficient for deducing the
existence of Galois extensions with such groups, as well as for the construction of polynomials generating such extensions. Nevertheless, the effective construction requires the
solution of a non-linear system of equations, a problem which is known to be very hard
from a complexity point of view. Thus, in practice, the computation of polynomials is
restricted to rather small degree, to the case of stem fields of genus zero and also to few
(mostly three) ramification points. For several applications, for example for the solution
of embedding problems, it is sometimes necessary to find Galois extensions of the rationals with given group and with complex conjugation lying in a prescribed conjugacy
class. But it is well known (see for example Malle and Matzat (1999), Ex. I.10.2) that
three point ramified Galois extensions almost never have totally real specializations, for
example.
In this paper we give 2-, 3- and 4-parameter polynomials for certain (mostly nonsolvable) groups which, from a certain point of view, correspond to Galois extensions
ramified in at least four points, with the property that these admit (infinitely many) totally real, Galois group preserving specializations. For example we obtain a two-parameter
polynomial for the sporadic simple Mathieu group M12 over Q. Suitable specializations
then yield totally real number fields with groups M11 and M12.
Acknowledgement: I would like to thank Peter M¨uller for very useful conversations on
the topic of this paper.
つづく
464(3): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/04(月)17:48 ID:Qu1TcOyQ(5/28) AAS
>>463
つづき
5. The group PGL2(7)
Theorem 5.1. The polynomial
f(a, t, X) := X(X^7 + X^6 + 14aX^5 + 7aX^4 + 49a^2X^3 + 14a^2X^2 + 49a^3X + 7a^3)+t(7X^2 + X + 1)
has Galois group PGL2(7) over Q(a, t). The branch cycle description with respect to t is
of type (2^3, 2^3, 2^3, 6).
Remark. The polynomial f in Theorem 5.1 has totally real specializations; for example,
if a = ?2 and ?1 ? t ? 6. One example of a totally real PGL2(7) polynomial is given by
X^8 ? 2 X^7 ? 35 X^6 + 308 X^4 + 308 X^3 ? 462 X^2 ? 556 X + 6.
The field extension with group L3(2) constructed by LaMacchia (1980) is embeddable
into a PGL2(7)-field. Since PGL2(7) does not have a faithful permutation representation of degree 7, a stem field of that Galois extension will have degree 8. A generating
polynomial is given by:
Theorem 5.2. The polynomial
f(a, t, X) := (X^2 ? 4 (16 a ? 3) b) (X^3 + 4 b X^2 ? 2 X (16 a ? 3) b ? 4 (16 a ? 3) b^2)2
? t(2X^5 + (8 a^2 ? 24 a ? 31)X^4 ? 4(40 a + 17)bX^3 ? 4(16 a ? 3)
(6 a^2 ? 20 a ? 29)bX^2
+ 32 (88 a^6 ? 497 a^5 ? 302 a^4 + 114 a^3 + 854 a^2 ? 9 a ? 18) X + 16 (64 a^8 ? 2672 a^7
+ 2906 a^6 + 550 a^5 + 3466 a^4 ? 118 a^3 ? 2731 a^2 + 366 a ? 90))
+ t2(X^2 ? 4 (2 a ? 1) X + 4 (31 a^2 ? 4 a + 1))
(where b := a^2 ? 2 a ? 1) has Galois group PGL2(7) over Q(a, t). The branch cycle
description with respect to t is of type (2^4, 2^4, 2^3, 6).
The polynomial f(a, t2, X) has Galois group L2(7) over Q(a, t), with branch cycle description of type (2^4, 2^4, 2^4, 2^4, 3^2) with respect to t.
(引用終り)
以上
465(1): 2019/11/04(月)18:02 ID:lsGvCqzx(6/24) AAS
>>460
＞小学生に教えているんだってな
いいえ、どこからそんな電波を受信したの？ｗ
>>461
「情報科学が専門」というのは事実
昔のことですがね
>>462-464
また、わかりもしない英文をコピペですか？
いったい、あなたは何がしたいの
もう、あなたの実力は露見しちゃってるから
マウンティングは無理ですよ
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