MadPidgeon's profile - activity

Thanks! I will look into it.

I ran the following code to factor a polynomial over a number field:

U.<z> = CyclotomicField(32)
P.<x> = PolynomialRing(U)
f = x^16-256
print f.factor()

This code works for all substitutions of 256 by another value I have tried, but this one gives an error:

Traceback (most recent call last):
File "p_is_2_test.sage.py", line 10, in <module>
print f.factor()
File "sage/rings/polynomial/polynomial_element.pyx", line 4199, in sage.rings.polynomial.polynomial_element.Polynomial.factor (build/cythonized/sage/rings/polynomial/polynomial_element.c:39418)
File "cypari2/auto_gen.pxi", line 17246, in cypari2.gen.Gen_auto.nffactor
File "cypari2/handle_error.pyx", line 196, in cypari2.handle_error._pari_err_handle
cypari2.handle_error.PariError: inconsistent concatenation t_COL (8 elts) , t_VEC (8 elts)

This seems like some bug in sage, but I am not quite sure what to make of the error. Does anyone know how to deal with this error and to properly let sage factor this polynomial?

Is there any way to coerce the formal composite map to one of the former type? I compute the product of O(n) morphisms as f = one and for phi in l: f = phi*f, and the latter becomes an unreadable block of composite maps when I print the result, and it doesn't help the algorithmic complexity either when trying to evaluate the function I presume. I can work around this by reverse iterating the list and doing right multiplication with phi instead, but this feels like a hack. Furthermore, my question was whether any piece of documentation justifies this counter intuitive result.

I have a number field U for which I consider its automorphisms through Hom(U,U). The identity one=Hom(U,U).identity() behaves weirdly under right multiplication:

U = CyclotomicField(3)
f = Hom(U,U)[1]
print f
print "--------------------"
one = Hom(U,U).identity()
print f*one
print "--------------------"
print one*f
print "--------------------"
print f*f

When I run this code, I expect f to be printed thrice, followed by the identity morphism. However, while the first and third output do in fact both print f, the second prints

Composite map:
  From: Cyclotomic Field of order 3 and degree 2
  To:   Cyclotomic Field of order 3 and degree 2
  Defn:   Identity endomorphism of Cyclotomic Field of order 3 and degree 2
        then
          Ring endomorphism of Cyclotomic Field of order 3 and degree 2
          Defn: zeta3 |--> -zeta3 - 1

Is this a bug or is this this behaviour explained somewhere in the documentation?

Maybe I should have included this, but at `print K.gens(), L.gens()' sage answers with

(gen1, zeta3) (gen2, zeta3)

so clearly K has two generators, I think? Secondly, how does it decide where to send zeta3? In this case there is little room, but if we take

P.<X> = PolynomialRing(QQ)
R.<s2> = QQ.extension(X^2-2)
K.<s3> = R.extension(X^2-3)
print K.gens()
H = K.hom( [-s3,-s2], K )
print H

this should also give a valid morphism instead of the one returned by K.hom([-s3]) that maps s2 to itself.

Given two number fields, I want to construct a morphism between them. For this I tried to use the hom member-function of the NumberField object as follows:

R.<zeta3> = CyclotomicField(3)
P.<X> = PolynomialRing(R)
K.<gen1> = R.extension(X^3-zeta3)
L.<gen2> = R.extension(X^3-zeta3^2)
print K.gens(), L.gens()
H = K.hom( [gen2,zeta3^2], L )
print H

The help page of hom specifies:

Return the unique homomorphism from self to codomain that sends self.gens() to the entries of im_gens. Raises a TypeError if there is no such homomorphism.

However, instead of TypeError, I get an incomprehensible error:

File "/home/sage/bin/sage2/local/lib/python2.7/site-packages/sage/rings/number_field/number_field.py", line 1670, in _element_constructor_ raise ValueError("Length must be equal to the degree of this number field") ValueError: Length must be equal to the degree of this number field

What am I doing wrong? Is there a better way to define this morphism?