#7097: bug in polynomial factorization over number fields
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Reporter: cremona | Owner: tbd
Type: defect | Status: new
Priority: major | Milestone: sage-4.1.3
Component: algebra | Keywords: polynomial factor root number field
Reviewer: | Author:
Merged: |
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Using 4.1.2.rc0.
Define a number field K with the irreducible polynomial g of degree 24,
and another polynomial f:
{{{
sage: x = polygen(QQ)
sage: f = 8*x^9 + 42*x^6 + 6*x^3 - 1
sage: g = x^24 - 12*x^23 + 72*x^22 - 286*x^21 + 849*x^20 - 2022*x^19 +
4034*x^18 - 6894*x^17 + 10182*x^16 - 13048*x^15 + 14532*x^14 - 13974*x^13
+ 11365*x^12 - 7578*x^11 + 4038*x^10 - 1766*x^9 + 762*x^8 - 408*x^7 +
236*x^6 - 126*x^5 + 69*x^4 - 38*x^3 + 18*x^2 - 6*x + 1
sage: assert g.is_irreducible()
sage: K.<a> = NumberField(g)
}}}
f has a root in K:
{{{
sage: x0 = 3260097/3158212*a^22 - 35861067/3158212*a^21 +
197810817/3158212*a^20 - 722970825/3158212*a^19 + 1980508347/3158212*a^18
- 4374189477/3158212*a^17 + 4059860553/1579106*a^16 -
6442403031/1579106*a^15 + 17542341771/3158212*a^14 -
20537782665/3158212*a^13 + 20658463789/3158212*a^12 -
17502836649/3158212*a^11 + 11908953451/3158212*a^10 -
6086953981/3158212*a^9 + 559822335/789553*a^8 - 194545353/789553*a^7 +
505969453/3158212*a^6 - 338959407/3158212*a^5 + 155204647/3158212*a^4 -
79628015/3158212*a^3 + 57339525/3158212*a^2 - 26692783/3158212*a +
1636338/789553
sage: f(x0)
0
}}}
(in fact f splits over K) but the root-finder does not find any:
{{{
sage: f.roots(K)
[]
}}}
What is worse, even though f factors over QQ:
{{{
sage: f.factor()
(8) * (x^3 + 1/4) * (x^6 + 5*x^3 - 1/2)
}}}
it apparently does not over K!
{{{
sage: f.change_ring(K).factor()
(64) * (x^9 + 21/4*x^6 + 3/4*x^3 - 1/8)
}}}
Remark: f is the 4-division polynomial of the elliptic curve [0,0,1,0,0]
and K is the 4-division field.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/7097>
Sage <http://www.sagemath.org>
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