22/12/25 09:30:49.45 4mPovfMa.net
>>34 追加補足
まず(参考)
URLﾘﾝｸ(www-users.york.ac.uk)
Symmetries of Equations: An Introduction
to Galois Theory
Brent Everitt 2007
Department of Mathematics, University of York,
P6
(1.9) If this was always the case, things would be very simple: Galois theory would just be the study
of the “shapes” formed by the roots of polynomials, and the symmetries of those shapes. It would be a
branch of planar geometry.
But things are not so simple. If we look at the solutions to x
5 - 2 = 0, something quite different
happens：
（図があるが略（というかここには示せない））
（言葉で書くと、複素平面上の半径r=α =2^1/5上に頂点を持つ正5角形で、頂点の一つが実数α =2^1/5で、そこから反時計回りに、αω,αω^2,αω^3,αω^4 と頂点が配置された図）
α =2^1/5
ω：1の5乗根
We will see later on how to obtain these expressions for the roots. A pentagon has 10 geometric symmetries, and you can check that all arise as symmetries of the roots of x^5 - 2 using the same reasoning as in
the previous example. But this reasoning also gives a symmetry that moves the vertices of the pentagon
according to:
（図があるが略（というかここには示せない））
（言葉で書くと、α は不動でαω→αω^3→αω^4→αω^2(→元のαωに戻る巡回置換の図）
This is not a geometrical symmetry! Later we will see that for p > 2 a prime number, the solutions to x^p - 2 = 0 have p(p - 1) symmetries.
(P7 Exercise 7 に、この部分が問題として出されている）
追記
余談だが、表紙のサッカーボールの図があり、表紙を開くとP2にこれを交代群A5のCaylayグラフにした見事な図示がある
これは、一見の価値ありです！
(引用終り)
つづく