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# Kummer Degrees
A SageMath script that computes the degrees of Kummer Extensions of the rational numbers.
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.
Columns correspond to values of M, rows to values of N
The degree of the Kummer extension (M,N) can be extracted by taking
the value f (failure) of the entry at (gcd(N,N0),gcd(M,M0)) and
simply computing ed(M,N) / f, where ed(M,N) is the expected degree
of the Kummer extension.
In this case (-1 is in G), we have ed(M,N) = 2^e*phi(M)*N^r,
where e=1 if N is even and e=0 if N is odd.
where r is the rank of G.
| 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 16 16 32
The following table shows the total failure of Kummer degrees in
case the quotient M/N is ODD.
This table can be read exactly as the first 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 8 8 8 16 16
```
## 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)
2304
```
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