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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])
+output:
+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.
+
+sage: KummerDegree([-36,12,-1],120,24)
+output:
+2304
+
+-------------------------------------------------------------------------------