谁能帮下载伽罗瓦理论天才的激情 [章璞 著] 2013年版.pdf
https://download.csdn.net/download/yulao2001/14992626伽罗瓦理论天才的激情 [章璞 著] 2013年版.pdf 已经上传到百度网盘. 链接: https://pan.baidu.com/s/1KfstnIDkkj7JNZ7mz8fL6g 提取码: yaws liangbch 发表于 2021-5-13 08:59
已经上传到百度网盘. 链接: https://pan.baidu.com/s/1KfstnIDkkj7JNZ7mz8fL6g 提取码: yaws
谢谢了，我已经把这个电子书传到论坛QQ群了 https://bbs.emath.ac.cn/thread-3018-1-1.html
这个五次方程怎么解？
求一个可解的五次方程
https://bbs.emath.ac.cn/thread-3024-1-1.html
y^5+10y^2-15y+6=0有根式解吗？
https://bbs.emath.ac.cn/thread-5946-1-1.html
看完这本书，估计这两个问题就能彻底解决了 为什么课本上计算Galois群都是先求出根再计算的（不可解的除外）？ - 知乎
https://www.zhihu.com/question/458961859/answer/1883098711
这个回答我觉得有意思 https://kconrad.math.uconn.edu/blurbs/galoistheory/galoisSnAn.pdf
RECOGNIZING GALOIS GROUPS Sn AND An
没怎么看懂 本帖最后由 mathematica 于 2021-5-14 14:21 编辑
//变量是t
P<t>:=PolynomialAlgebra(Rationals());
//注意多项式变量是t,下面方程不可以求根式解
f:=1*t^5+10*t^2-15*t+6;
f:=f*(t^2+3*t+2);
//注意多项式变量是t,下面方程可以求根式解
f:=32*t^5+3349456*t^4-5941616812296*t^3-585145514845851080*t^2+147013447513276833423286*t+15377302441624829616294559439;
//注意多项式变量是t,下面方程可以求根式解
f:=t^8-t^7+29*t^2+29;
G:=GaloisGroup(f);
print G;
//看群是否可解
IsSolvable(G);
网站：
http://magma.maths.usyd.edu.au/calc/
运行结果：
Permutation group G acting on a set of cardinality 8
Order = 56 = 2^3 * 7
(1, 4, 6, 2, 5, 3, 8)
(1, 4)(2, 8)(3, 7)(5, 6)
(1, 3)(2, 5)(4, 7)(6, 8)
(1, 5)(2, 3)(4, 6)(7, 8)
true
参考资料
Galois group command for Magma online calculator?
https://math.stackexchange.com/questions/642329/galois-group-command-for-magma-online-calculator?noredirect=1
On the solvable octic x8−x7+29x2+29=0
https://mathoverflow.net/questions/145145/on-the-solvable-octic-x8-x729x229-0
Galois Groups
Finding Galois groups (of normal closures) of algebraic fields is a hard problem, in general. All practical currently-used algorithms fall into two groups: The absolute resolvent method and the method of Stauduhar .
The Magma implementation is based on an extension of the method of Stauduhar by , and recent work by Klüners and Fieker , Elsenhans , , and Sutherland .
For polynomials over Z, Q, number fields and global function fields and irreducible polynomials over function fields over Q, Magma is able to compute the Galois group without any a-priori restrictions on the degree. Note, however, that the running time and memory constraints can make computations in degree >50 impossible, although computations in degree >200 have been successful as well. In contrast to the absolute resolvent method, it also provides the explicit action on the roots of the polynomial f which generates the algebraic field. On demand, the older version which is restricted to a maximum degree of 23, is still available.
Roughly speaking, the method of Stauduhar traverses the subgroup lattice of transitive permutation groups of degree n from the symmetric group to the actual Galois group. This is done by using so-called relative resolvents. Resolvents are polynomials whose splitting fields are subfields of the splitting field of the given polynomial which are computed using approximations of the roots of the polynomial f.
If the field (or the field defined by a polynomial) has subfields (i.e. the Galois group is imprimitive) the current implementation changes the starting point of the algorithm in the subgroup lattice, to get as close as possible to the actual Galois group. This is done via computation of subfields of a stem field of f, that is the field extension of Q which we get by adjoining a root of f to Q. Using this knowledge of the subfields, the Galois group is found as a subgroup of the intersection of suitable wreath products which may be easily computed. This intersection is a good starting point for the algorithm.
If the field (or the field defined by a polynomial) does not have subfields (i.e. the Galois group is primitive) we use a combination of the method of Stauduhar and the absolute resolvent method. The Frobenius automorphism of the underlying p-adic field or the complex conjugation, when using complex approximations of the roots of the polynomial f, already determines a subgroup of the Galois group, which is used to speed up computations in the primitive case.
http://magma.maths.usyd.edu.au/magma/handbook/text/419 Galois Group Computation for Rational Polynomials
https://core.ac.uk/download/pdf/82514243.pdf
https://www.math.colostate.edu/~hulpke/talks/galoistalk.pdf
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