#!/usr/bin/env spython import sys from sage.all import * from random import sample from collections import Counter def t(n,k): return binomial(n-floor((k+1)/2),floor(k/2)) def alpha(n,k): return (-1)**(k*(k-1)/2) def pbin(n): return sum([alpha(n,k)*t(n,k)*x**(n-k) for k in range(n+1)]) def genCyclicPoly(n): x = var('x') k = 0 while True: p = k*n+1 k += 1 if is_prime(p): break y = primitive_root(p) K = CyclotomicField(p,names='a') a = K.gen() o = Mod(y**n,p).multiplicative_order() s = sum([a**((y**n)**i%p) for i in range(o)]) pol = s.minpoly('x') return (pol,k-1,p,y) for n in range(1,40+1): pol,k,p,y = genCyclicPoly(n) print n print pol printFor example for n=17−12=8 we get the following polynomial: x8+x7−7x6−6x5+15x4+10x3−10x2−4x+1 Looking at the coefficients of the polynomial and searching for them in OEIS we find the sequence A065941. Consider the following polynomial: p(n,x)=n∑k=0(n−⌊k+12⌋⌊k2⌋)⋅(−1)k(k+1)2⋅xn−k We conjecture that the following is true (let p>2 be prime): Gal(p(n,x))=Cn⇔n=p−12⇔p(n,x) is irreducible If someone finds a proof or a counterexample to this, please let me know.
Mittwoch, 30. September 2015
Computing polynomials with cyclic Galois groups in SAGE
Here is some SAGE code to compute a polynomial with degree n which has cyclic Galois group.
The computation is done as in the example
https://en.wikipedia.org/wiki/Inverse_Galois_problem#Worked_example:_the_cyclic_group_of_order_three
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