commit 3b815435cbb3119ccc55063cee595b0bf4e213cb
parent 0add0f16c11a2c77e474aee0731265f49c13203b
Author: Sebastiano Tronto <sebastiano.tronto@gmail.com>
Date: Fri, 20 Sep 2019 11:41:53 +0200
New documentation
Diffstat:
| M | README.md | | | 60 | ++++++++++++++++++++++++++++++++++++++++++++++++++++++------ |
1 file changed, 54 insertions(+), 6 deletions(-)
diff --git a/README.md b/README.md
@@ -61,18 +61,66 @@ sage: KummerDegree([144,27,49/81,-1/125,121/13],36*10^6,36*10^6)
## 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.
+Outputs the description of the failure of maximality for all possible values
+of M and N.
-Example:
+INPUT:
+G - a list of generators for the group G
+
+OUPUT:
+The first part of the output consist of two positive integers M_0 and N_0.
+N_0 is always a divisor of M_0.
+
+The second part of the output can be either one or two tables, depending on
+the group G. In case -1 is not an element of G, there is only one table,
+otherwise two.
+
+In case -1 does not belong to G:
+The rows of the table(s) are labelled with divisors of N_0, while the columns
+with divisors of M_0. The total failure of maximality of the Kummer Extension
+Q_{M,N}, i.e. the ratio between phi(M)*N^rank(G) and the degree of Q_{M,N}
+over Q, is given by the (i,j)-th entry of the table for i=gcd(N,N_0) and
+j=gcd(M,M_0).
+
+In case -1 belongs to G, we need to distinguish two cases, depending on the
+parity of M/N. The two tables are similar to the case described
+above, and can be seen in the example below.
+
+The output includes an explanation of the tables and how to read them to
+deduce the actual degrees (that can be computed with KummerDegree).
+
+EXAMPLES:
```
+sage: TotalKummerFailure([-36,12])
+M_0 = 24
+N_0 = 8
+The following table shows the total failure of Kummer degrees.
+The degree of the Kummer extension (M,N) is e / f, where e = phi(M)*N^rank(G)
+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 | 1 1 1 2 2 2 4 4
+ 4 | 4 4 4 4 4 4 8 8
+ 8 | 4 4 4 4 4 4 4 8
+
+```
+
+```
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).
+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
- - - - - - - - - -
@@ -81,7 +129,8 @@ The degree of the Kummer extension (M,N) is e / f, where e = phi(M)*N^rank(G) if
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.
+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
- - - - - - - - - -
@@ -92,4 +141,3 @@ The following table shows the total failure of Kummer degrees if the quotient M/
```
-
