ASKSAGE: Sage Q&A Forum - Latest question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 21 Jan 2020 09:04:23 -0600.is_galois Computationhttp://ask.sagemath.org/question/49620/is_galois-computation/Given an irreducible polynomial $f$, Sage computes whether a given field $K= \mathbb{Q}(f)$ is Galois with $K$.is_galois. This works well if $f$ is of low degree, say 1-20. But when $f$ is large, say degree 100 or more, this is very time consuming.
For $K$ to be Galois, it must have the same degree as $f$ and because we would expect (at random) $f$ to have Galois group $S_n$, $\text{Gal}(K/\mathbb{Q})$ will be very large. So in theory, determining 'Is Galois Y/N' should run much faster than actually computing the Galois group - which is very hard.
How does Sage .is_galois work? Does it try to compute the Galois group and compare sizes, or does it use some other method? If in computing $\text{Gal}(K/\mathbb{Q})$ you find a group with size at least $> \deg f$, does it automatically stop and give 'False'? If not, is there a way to force such a feature using features already built into Sage?nmbthrTue, 21 Jan 2020 09:04:23 -0600http://ask.sagemath.org/question/49620/implement given divisor as a specific hyperelliptic curve divisorhttp://ask.sagemath.org/question/40589/implement-given-divisor-as-a-specific-hyperelliptic-curve-divisor/I get the values of a divisor for a hyperelliptic curve from a research paper to ensure the divisor is correct and i want to use this divisor on calculation.
so Idid the following:
[A Secured Cloud System using Hyper Elliptic Curve](https://www.google.com.eg/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwjLjIvt_cvYAhWGvBQKHWlACL8QFggqMAE&url=https%3A%2F%2Fwww.ijser.org%2Fresearchpaper%2FA-Secured-Cloud-System-using-Hyper-Elliptic-Curve-Cryptography.pdf&usg=AOvVaw3gkCA7QngzSue-BZCV2DB8)
and in sage:
p = 4112543547855339322343814790708185367671872426434747235319998473455582535888229747778325047393413053
K = GF(p)
R.<x> = K[]
f = x^5 + 7943193*x^4 + 6521255*x^3 + 1065528*x^2 + 3279922*x + 3728927
C = HyperellipticCurve( f )
J = C.jacobian()
X = J(K)
u, v = x^2 + 22457213658579645161*x + 62960708771725664757, 65279057408798633572*x + 32004384923913711271
D = X( [u,v] )
> verbose 0 (3324:
> multi_polynomial_ideal.py,
> groebner_basis) Warning: falling back
> to very slow toy implementation.
> verbose 0 (1083:
> multi_polynomial_ideal.py, dimension)
> Warning: falling back to very slow toy
> implementation. Error in lines 9-9
> Traceback (most recent call last):
> File
> "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py",
> line 1013, in execute
> exec compile(block+'\n', '', 'single') in namespace, locals File
> "", line 1, in <module> File
> "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/schemes/hyperelliptic_curves/jacobian_homset.py",
> line 145, in __call__
> return JacobianMorphism_divisor_class_field(self,
> tuple(P)) File
> "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/schemes/hyperelliptic_curves/jacobian_morphism.py",
> line 388, in __init__
> polys, C)) ValueError: Argument polys (= (x^2 + 22457213658579645161*x
> + 62960708771725664757, 65279057408798633572*x +
> 32004384923913711271)) must be divisor
> on curve Hyperelliptic Curve over
> Finite Field of size
> 4112543547855339322343814790708185367671872426434747235319998473455582535888229747778325047393413053
> defined by y^2 = x^5 + 7943193*x^4 +
> 6521255*x^3 + 1065528*x^2 + 3279922*x
> + 3728927.
Does this mean the paper is wrong or I am at wrong.
and if so, how can I modify my work?sherifasagewadSat, 13 Jan 2018 05:03:30 -0600http://ask.sagemath.org/question/40589/What is a PARI group?http://ask.sagemath.org/question/40369/what-is-a-pari-group/Take for instance the example:
G = NumberField(x^5 + 15*x + 12, 't').galois_group(type="pari"); G
This gives an output:
Galois group PARI group [20, -1, 3, "F(5) = 5:4"] of degree 5 of the Number Field in t with defining polynomial x^5 + 15*x + 12
So I wonder what is the meaning of [20, -1, 3, "F(5) = 5:4"]. Is it some kind of permutation?
The relevant page of the [documentation](http://doc.sagemath.org/html/en/reference/groups/sage/groups/pari_group.html?highlight=pari%20group#module-sage.groups.pari_group) doesn't clarify it.Rodrigo RayaSat, 30 Dec 2017 06:30:00 -0600http://ask.sagemath.org/question/40369/Computing Galois group of a polynomial?http://ask.sagemath.org/question/24289/computing-galois-group-of-a-polynomial/ Here's my attempt so far:
x = polygen(QQ, 'x');
K.<z> = NumberField(x^4+x^2+2*x+1)
G = K.galois_group(type='gap')
And now I get several pages of error messages beginning with
verbose 0 (1780: permgroup_named.py, cardinality) Warning: TransitiveGroups requires the GAP database package. Please install it with ``sage -i database_gap``.
However! - I have installed this:
install_package()
[...
'database_gap-4.7.4',
...]
Anybody got any ideas on where I can go from here?AlasdairFri, 26 Sep 2014 01:41:20 -0500http://ask.sagemath.org/question/24289/question about GaloisGroup_v1?http://ask.sagemath.org/question/10845/question-about-galoisgroup_v1/Q1.<a> = NumberField(x^3- 2);Q1;
Q2.<b>= CyclotomicField(3);Q2;
Q3.<c> = Q2.extension(x^3 - 2);Q3;
Q3.absolute_polynomial();
G=Q3.galois_group();G;
G.number_field();Q
1.is_galois();
Q2.is_galois();
G2=Q2.galois_group('pari');G2;
G1.<m>=Q1.galois_group('pari');G1;
Traceback (click to the left of this block for traceback)
...
AttributeError: 'GaloisGroup_v1' object has no attribute '_first_ngens'
cjshWed, 18 Dec 2013 21:15:22 -0600http://ask.sagemath.org/question/10845/