def list_to_factored_poly_otherorder(s, galois=False):
""" Either return the polynomial in a nice factored form,
or return a pair, with first entry the factored polynomial
and the second entry a list describing the Galois groups
of the factors.
"""
gal_list = []
if len(s) == 1:
if galois:
return [str(s[0]), [[0, 0]]]
return str(s[0])
sfacts = factor(PolynomialRing(ZZ, 'T')(s))
sfacts_fc = [[v[0], v[1]] for v in sfacts]
if sfacts.unit() == -1:
sfacts_fc[0][0] *= -1
outstr = ''
x = var('x')
for v in sfacts_fc:
this_poly = v[0]
# if the factor is -1+T^2, replace it by 1-T^2
# this should happen an even number of times, mod powers
if this_poly.substitute(T=0) == -1:
this_poly = -1 * this_poly
v[0] = this_poly
if galois:
this_degree = this_poly.degree()
# hack because currently sage only handles monic polynomials:
this_poly = expand(x**this_degree * this_poly.substitute(T=1 / x))
this_number_field = NumberField(this_poly, "a")
this_gal = this_number_field.galois_group(type='pari')
this_t_number = this_gal.group()._pari_()[2]._sage_()
gal_list.append([this_degree, this_t_number])
vcf = v[0].list()
started = False
if len(sfacts) > 1 or v[1] > 1:
outstr += '('
for i in range(len(vcf)):
if vcf[i] != 0:
if started and vcf[i] > 0:
outstr += '+'
started = True
if i == 0:
outstr += str(vcf[i])
else:
if abs(vcf[i]) != 1:
outstr += str(vcf[i])
elif vcf[i] == -1:
outstr += '-'
if i == 1:
outstr += 'T'
elif i > 1:
outstr += 'T^{' + str(i) + '}'
if len(sfacts) > 1 or v[1] > 1:
outstr += ')'
if v[1] > 1:
outstr += '^{' + str(v[1]) + '}'
if galois:
if galois and len(sfacts_fc) == 2:
if sfacts[0][0].degree() == 2 and sfacts[1][0].degree() == 2:
troubletest = sfacts[0][0].disc() * sfacts[1][0].disc()
if troubletest.is_square():
gal_list = [[2, 1]]
return [outstr, gal_list]
return outstr
def list_to_factored_poly_otherorder(s, galois=False, vari = 'T'):
""" Either return the polynomial in a nice factored form,
or return a pair, with first entry the factored polynomial
and the second entry a list describing the Galois groups
of the factors.
vari allows to choose the variable of the polynomial to be returned.
"""
gal_list=[]
if len(s) == 1:
if galois:
return [str(s[0]), [[0,0]]]
return str(s[0])
sfacts = factor(PolynomialRing(ZZ, 'T')(s))
sfacts_fc = [[v[0],v[1]] for v in sfacts]
if sfacts.unit() == -1:
sfacts_fc[0][0] *= -1
outstr = ''
x = var('x')
for v in sfacts_fc:
this_poly = v[0]
# if the factor is -1+T^2, replace it by 1-T^2
# this should happen an even number of times, mod powers
if this_poly.substitute(T=0) == -1:
this_poly = -1*this_poly
v[0] = this_poly
if galois:
this_degree = this_poly.degree()
# hack because currently sage only handles monic polynomials:
this_poly = expand(x**this_degree*this_poly.substitute(T=1/x))
this_number_field = NumberField(this_poly, "a")
this_gal = this_number_field.galois_group(type='pari')
this_t_number = this_gal.group().__pari__()[2].sage()
gal_list.append([this_degree, this_t_number])
vcf = v[0].list()
started = False
if len(sfacts) > 1 or v[1] > 1:
outstr += '('
for i in range(len(vcf)):
if vcf[i] != 0:
if started and vcf[i] > 0:
outstr += '+'
started = True
if i == 0:
outstr += str(vcf[i])
else:
if abs(vcf[i]) != 1:
outstr += str(vcf[i])
elif vcf[i] == -1:
outstr += '-'
if i == 1:
outstr += vari #instead of putting in T for the variable, put in a variable of your choice
elif i > 1:
outstr += vari + '^{' + str(i) + '}'
if len(sfacts) > 1 or v[1] > 1:
outstr += ')'
if v[1] > 1:
outstr += '^{' + str(v[1]) + '}'
if galois:
if galois and len(sfacts_fc)==2:
if sfacts[0][0].degree()==2 and sfacts[1][0].degree()==2:
troubletest = sfacts[0][0].disc()*sfacts[1][0].disc()
if troubletest.is_square():
gal_list=[[2,1]]
return [outstr, gal_list]
return outstr