1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
|
# Kummer Degrees
A SageMath script that computes the degrees of Kummer Extensions of the
rational numbers.
A Kummer Extension of Q is a field extension of the form Q_{M,N}:=
Q(\zeta_M,G^{1/N}), where:
* M and N are integers with N dividing M;
* \zeta_M is a root of unity of order M;
* G is a finitely generated subgroup of the multiplicative group of Q;
* G^{1/N} is the set of all elements x of an algebraic closure of Q such that
x^n belongs to G.
It contains the following useful functions:
## TotalKummerFailure( G )
Outputs the description of the failure of maximality for all possible values of M and N. Here G is given as a list of generators (not necessarily a basis). G can also contain torsion. If G = <-1>, the program stops immediately.
Example:
```
sage: TotalKummerFailure([-36,12,-1])
M_0 = 24
N_0 = 8
The following table shows the total failure of Kummer degrees in case the quotient M/N is EVEN.
The degree of the Kummer extension (M,N) is e / f, where e = phi(M)*N^rank(G) if N is odd and e = 2*phi(M)*N^rank(G) if N is even and f is the entry of the table below at the row labelled with gcd(N,N0) and the column labelled with gcd(M,M0).
| 1 2 3 4 6 8 12 24
- - - - - - - - - -
1 | 1 1 1 1 1 1 1 1
2 | 4 4 4 4 4 4 8 8
4 | 4 4 4 4 4 8 8 16
8 | 8 8 8 8 8 8 8 16
The following table shows the total failure of Kummer degrees if the quotient M/N is ODD and is read as the previous one.
| 1 2 3 4 6 8 12 24
- - - - - - - - - -
1 | 1 1 1 1 1 1 1 1
2 | 2 2 2 2 4 2 4 4
4 | 2 2 2 4 4 4 8 8
8 | 4 4 4 4 4 4 8 8
```
## KummerDegree( G, M, N )
Returns the degree of the Kummer extension Q_{M,N}. Again, G is given simply as a list of generators and it may contain torsion.
Example:
```
sage: KummerDegree([-36,12,-1],120,24)
4608
```
|
