commit d794bad3364a7bf0c5ca12241d36a9ebb48c68d8
parent 4e06fe16c1a054dbbd981dc8b27e61a4cdb0cb10
Author: Sebastiano Tronto <sebastiano.tronto@gmail.com>
Date: Fri, 19 Apr 2019 11:38:07 +0200
Added instructions to README.md
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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
+
+-------------------------------------------------------------------------------
