ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 18 Sep 2019 01:19:57 +0200Elliptic Curve over finite fields - Irreducibility proofhttps://ask.sagemath.org/question/36115/elliptic-curve-over-finite-fields-irreducibility-proof/ Hello,
I am new to Sage and I want to prove an elliptic curve is irreducible.
How can I prove that?
Example:
- E = EllipticCurve(GF(256),[8,4,3,2,0])
Thank youemrealparslanWed, 28 Dec 2016 21:30:26 +0100https://ask.sagemath.org/question/36115/How sage checks the irreducibility of a polynomial?https://ask.sagemath.org/question/47951/how-sage-checks-the-irreducibility-of-a-polynomial/ Hi, given a polynomial we can check whether its irreducibility via `.is_irreducible()` command. I wonder how sage checks it so fast even though the polynomial has large degree with large coefficients?captainWed, 18 Sep 2019 01:19:57 +0200https://ask.sagemath.org/question/47951/Towered extension fields through chosen polynomialhttps://ask.sagemath.org/question/46862/towered-extension-fields-through-chosen-polynomial/(Edited, since `<u>` compiles in HTML as an underlyning tag, dan.)
K.<u> = GF(q)
KT.<u> = K.extension(2)
KTT.<u> = K.extension(6)
In this case `KTT` is a degree 12 extension. My question is:
Can I obtain `KTT` by `KTT.<u> = K.extension(x^6 - a)`, where `x^6 -a` is some irreducible polynomial over `KT` ???
Documentation gives nothing about this towering of fields.pchSat, 08 Jun 2019 14:23:35 +0200https://ask.sagemath.org/question/46862/Defining a 'nice' Compositumhttps://ask.sagemath.org/question/46314/defining-a-nice-compositum/ I'm having two difficulties which I assume are simple to resolve. I have a set field, say $K= \mathbb{Q}(x^2+1)$, and a set of polynomials I want to check if they are irreducible over $K$. If they are irreducible, I would like to form the compositum of $K$ and this field generated by $f$ and then find a 'nice' generator. For example,
K.<root> = NumberField(x^2+1);
R = K['x'];
poly = [x^2 + 1, x^2 + 2, x^2 + x + 1];
f = R(poly[0]);
if f.is_irreducible:
L = NumberField([x^2+1, f]);
There are two issues:
1. Even when f.is_irreducible() gives True, the polynomial is not always irreducible, as in the case above, so that the construction of L gives an error.
2. Even if L can be formed, how do I find a 'nice' generator for L, i.e. a single polynomial $g$ which generates L so that I can form a 'better' (in terms of computation speed) field $M= \mathbb{Q}(g)$?nmbthrTue, 23 Apr 2019 07:07:29 +0200https://ask.sagemath.org/question/46314/Defining Polynomial from Importhttps://ask.sagemath.org/question/46282/defining-polynomial-from-import/I am importing polynomials from LMFBD. So I have a set called data, which contains a polynomial in certain entries. So for example, data[0][0] may be a polynomial $x^2+x+1$. I want to check if this polynomial is irreducible over some number field $K$ I have defined. So I tried something like...
K = NumberField(y^2+1);
f = data[0][0];
f.change_ring(K);
f.is_irreducible()
But I get the error 'sage.symbolic.expression.Expression' object has no attribute 'is_irreducible'. How would I do this?
nmbthrSat, 20 Apr 2019 06:16:46 +0200https://ask.sagemath.org/question/46282/Decompose polynomial by other irreducible polynomialhttps://ask.sagemath.org/question/25530/decompose-polynomial-by-other-irreducible-polynomial/Suppose I have irreducible polynomial $v(x)$ over $\mathbb Q$ (or arbitrary field). I want to decompose any other $f(x) \in \mathbb Q[x]$ by powers of $v$. Like this
$$f(x)=a_n(x)(v(x))^n + \dots + a_1(x)v(x) + a_0(x)$$
Is there some fast(?) way to do it in Sage except by hand writing your own function?
UPD. I forget to add that $\deg a_i < \deg v$.
UPD2. Naive solution:
degree = f.degree()//v.degree()
decomposition = [None] * (degree + 1)
for i in range(degree+1):
decomposition[i] = f%v
f //= v
return decomposition
But may be there is some native solutionpetRUShkaThu, 15 Jan 2015 13:54:09 +0100https://ask.sagemath.org/question/25530/