Yes. That is strange for me. Since it would be easier to compute the
factorization of an ideal generate by a prime.
For example: Using the Proposition that asserts that if K is a number field and
the ring of integers is of the form O_K=Z[alpha] (where f(t) is the minimal
polynomial over Z of alpha) then to compute the factorization of the ideal
<p>O_K where p is a rational prime is enough to factorize f(t)mod p. That is, if
f(t)=f_1(t)^r_1...f_s(t)^r_s (mod p) then the factorization on prime ideal is of
the form
<p>O_K=<p,f_1(alpha)>^r_1...<p,f_s(alpha)>^r_s.
Enrique
>
> But why would Sage be computing the class group in order to factor 2 in K?
>
> John
>
> On 29/12/2007, [EMAIL PROTECTED]
> <[EMAIL PROTECTED]> wrote:
> >
> > Hello:
> >
> > I am working at my Number Theory lectures and I have found a bug (?). This
> > is
> > the output:
> >
> > /////////////////// SAGE 2.9.1 ///////////////////
> > sage: K.<a>=CyclotomicField(23)
> > sage: O=K.maximal_order()
> > sage: (2*O).factor()
> > *** Warning: large Minkowski bound: certification will be VERY long.
> > Traceback (most recent call last):
> > File "<stdin>", line 1, in <module>
> > File "/home/notebook/sage_notebook/worksheets/admin/3/code/13.py",
> > line 4, in <module>
> > exec compile(ur'(Integer(2)*O).factor()' + '\n', '', 'single')
> > File
> > "/usr/local/sage/local/lib/python2.5/site-packages/sympy/plotting/",
> > line 1, in <module>
> >
> > File "sage_object.pyx", line 92, in
> > sage.structure.sage_object.SageObject.__repr__
> > File
> > "/usr/local/sage/local/lib/python2.5/site-packages/sage/structure/factor\
> > ization.py", line 187, in _repr_
> > t = str(self[i][0])
> > File
> > "/usr/local/sage/local/lib/python2.5/site-packages/sage/rings/number_fie\
> > ld/number_field_ideal.py", line 218, in __repr__
> > return "Fractional ideal %s"%self._repr_short()
> > File
> > "/usr/local/sage/local/lib/python2.5/site-packages/sage/rings/number_fie\
> > ld/number_field_ideal.py", line 235, in _repr_short
> > return '(%s)'%(', '.join([str(x) for x in self.gens_reduced()]))
> > File
> > "/usr/local/sage/local/lib/python2.5/site-packages/sage/rings/number_fie\
> > ld/number_field_ideal.py", line 553, in gens_reduced
> > dummy = self.is_principal(proof)
> > File
> > "/usr/local/sage/local/lib/python2.5/site-packages/sage/rings/number_fie\
> > ld/number_field_ideal.py", line 714, in is_principal
> > bnf = self.number_field().pari_bnf(proof)
> > File
> > "/usr/local/sage/local/lib/python2.5/site-packages/sage/rings/number_fie\
> > ld/number_field.py", line 1464, in pari_bnf
> > self.pari_bnf_certify()
> > File
> > "/usr/local/sage/local/lib/python2.5/site-packages/sage/rings/number_fie\
> > ld/number_field.py", line 1497, in pari_bnf_certify
> > if self.pari_bnf(certify=False, units=True).bnfcertify() != 1:
> > File "gen.pyx", line 6474, in sage.libs.pari.gen._pari_trap
> > sage.libs.pari.gen.PariError: not enough precomputed primes, need
> > primelimit ~ (35)
> >
> > But if you type the following lines using gp interface, it works:
> >
> > sage: K=gp.bnfinit(cyclotomic_polynomial(23))
> > sage: gp.idealfactor(K,2)
> >
> > [[2, [1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
> > 0]~, 1, 11, [1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
> > 0, 0]~], 1; [2, [1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0,
> > 0, 0, 0, 0]~, 1, 11, [1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0,
> > 0, 0, 0, 0, 0]~], 1]
> >
> >
> > All the best,
> >
> > Enrique
> >
> >
> > --------------------------------------------------------------------------
> > Mensaje enviado mediante una herramienta Webmail integrada en *El Rincon*:
> > ------------->>>>>>>> https://rincon.uam.es <<<<<<<<--------------
> >
> >
> >
> > >
> >
>
>
> --
> John Cremona
>
> >
>
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