From 31f855d9a13e237e81c46d1c363a9c74e443af43 Mon Sep 17 00:00:00 2001 From: Sebastiano Tronto Date: Tue, 17 Sep 2019 10:26:45 +0200 Subject: Removed some commented code, changed README.md --- README.md | 38 ++++++++++++++------------------------ 1 file changed, 14 insertions(+), 24 deletions(-) (limited to 'README.md') diff --git a/README.md b/README.md index 94ad39d..5fbdb34 100644 --- a/README.md +++ b/README.md @@ -9,39 +9,29 @@ Outputs the description of the failure of maximality for all possible values of 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. +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 8 8 8 16 16 + 8 | 4 4 4 4 4 4 8 8 ``` @@ -52,5 +42,5 @@ Returns the degree of the Kummer extension Q_{M,N}. Again, G is given simply as Example: ``` sage: KummerDegree([-36,12,-1],120,24) -2304 +4608 ``` -- cgit v1.3