Hi,
This computation takes long time but Sage is able to do it. Doing it
step by step, you can observe the relation between the degree of the
field and the time needed to generate it.
sage: a=2
sage: b=-1
sage: c=5
sage: ra = QQbar(a).sqrt(); ra.exactify()
sage: rb = QQbar(b).sqrt(); rb.exactify()
sage: rc = QQbar(c).sqrt(); rc.exactify()
sage: K, gens, phi= number_field_elements_from_algebraics(
....: [ra,rb,rc], minimal=True)
sage: g = phi(K.gen())
sage: alpha= 1 + ra
sage: gamma= 2 + rc
sage: A= (alpha^2 * gamma)/ ( (alpha+gamma) * (alpha*gamma + b) )
sage: A.exactify()
sage: C= (alpha+gamma)*(alpha*gamma + b) / (b*alpha)
sage: C.exactify()
sage: rA = A.sqrt()
sage: rA.exactify()
sage: rC = C.sqrt()
sage: rC.exactify()
sage: K2,gens2,phi2 = number_field_elements_from_algebraics(
....: [rA,rC], minimal=True)
sage: g2 = phi2(K2.gen())
sage: K3,gens3,phi3 = number_field_elements_from_algebraics(
....: [g,g2], minimal=True)
sage: g3 = phi3(K3.gen())
sage: delta= (alpha+gamma)/alpha
sage: delta.exactify()
sage: rdelta = delta.sqrt()
sage: rdelta.exactify()
sage: %time K4,gens4,phi4 = number_field_elements_from_algebraics(
....: [g3,rdelta], minimal=True)
sage: K4
Number Field in a with defining polynomial y^64 - 32*y^63 + 536*y^62 -
6160*y^61 + 53904*y^60 - 379232*y^59 + 2220272*y^58 - 11096704*y^57 +
48371100*y^56 - 187534480*y^55 + 658036744*y^54 - 2118167168*y^53 +
6310029040*y^52 - 17496563168*y^51 + 45354462568*y^50 -
110086558656*y^49 + 249092068530*y^48 - 518760267056*y^47 +
970049745040*y^46 - 1540658021296*y^45 + 1726215748448*y^44 +
261617504256*y^43 - 9416895523976*y^42 + 37922732526336*y^41 -
112535439872616*y^40 + 286729191976224*y^39 - 657670378395560*y^38 +
1379644802117040*y^37 - 2644268269215664*y^36 + 4577567236779360*y^35 -
6965533209709168*y^34 + 8539207658489472*y^33 - 5319331244483781*y^32 -
12389886738998320*y^31 + 65118197751618896*y^30 -
193580564430962864*y^29 + 472080249239522272*y^28 -
1019160749338571136*y^27 + 2001763007703523512*y^26 -
3634327413042005888*y^25 + 6153857939241808968*y^24 -
9737320234062776416*y^23 + 14333546151266502616*y^22 -
19364810168265444560*y^21 + 23281715700145502160*y^20 -
23008604561491319328*y^19 + 13767163228246461264*y^18 +
8765359686368418944*y^17 - 43200366094147621118*y^16 +
77999223597440406896*y^15 - 90519472303245161928*y^14 +
54028667281044557920*y^13 + 41367302912532250064*y^12 -
168969147746602154528*y^11 + 261672020286578277832*y^10 -
244896792544276915584*y^9 + 104844130048796679732*y^8 +
105596980415747506608*y^7 - 299690125602975518832*y^6 +
385834615403528683056*y^5 - 319833171820701032864*y^4 +
76406760284299949376*y^3 + 197175416506143879144*y^2 -
277360083651780693056*y + 182754695501929110289
On 10/11/15 08:35, Pierre wrote:
Hi,
I'm trying to construct a certain number field, of degree 64 over QQ (well,
I'd like to check that using Sage !).
It is constructed by adding a certain number of square roots. I have first
tried
F0= QQ
F1.<foo1>= F0.extension( polygen(F0)^2 - 2) ## adding sqrt(2)
F2.<foo2>= F1.extension( polygen(F1)^2 - 5) ## adding sqrt(2)
etc...
but each step was slower than the previous one, and the last step just
stalled, apparently. (Below are the details of the roots I'd like to add).
I have tried alternatively to use QQbar, which made the input very easy,
here is the whole thing:
a= 2
b= -1
c= 5
ra= QQbar(sqrt(a))
rc= QQbar(sqrt(c))
rb= QQbar(sqrt(b))
alpha= 1 + ra
gamma= 2 + rc
A= (alpha^2 * gamma)/ ( (alpha+gamma) * (alpha*gamma + b) )
C= (alpha+gamma)*(alpha*gamma + b) / (b*alpha)
delta= (alpha+gamma)/alpha
rA= sqrt(A)
rC= sqrt(C)
rdelta= sqrt(delta)
## and finally:
K, gens, phi= number_field_elements_from_algebraics([ra, rb, rc, rA, rC,
rdelta], minimal= True)
However this last command takes forever. (This is on SMC.)
Is there anything else that I could try?
Thanks !
Pierre
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