Ewan Delanoy on Thu, 24 Jan 2013 09:28:38 +0100

 Field inclusion problem

Hello all,

I would like to have the PARI-GP expertsâs advice about the feasibility of the following computation.
I have two polynomials A and B, given below, with A of degree 9 and B of degree 72. I know that if b is a root of B, then Q(b) is the decomposition field of A, so that there are three polynomials A_1,A_2,A_3 in b, each of degree at most 72,  such that A factorizes as (X-A1(b))(X-A2(b))(X-A3(b)).
The goal is to compute exactly the (ugly & complicated) coefficients of A1,A2,A3. I have tried to compute the roots of A and B to a large precision, then use lindep. Unfortunately, even when I set \p to 1000, I still get false coefficients.
Would that be considered a reasonable computation in PARi-GP ? Perhaps there are other methods than using lindep ?
Note that the B polynomial is quite complicated, Iâd be quite happy to replace it with a simpler polynomial that still corresponds to the decomposition field of A.

Thanks in advance for any help,

Ewan

polynomial_called_a=64*a^9 + 192*a^7 + 240*a^6 + 36*a^5 + 552*a^4 - 847*a^3 - 540*a^2 + 273*a + 143

polynomial_called_b=81474976710656*b^72 + 999799117276250112*b^70 + 1870401220242309120*b^69 + 1587204320361625681920*b^68 + 6441512761514344316928*b^67 + 1484884374331600352051200*b^66 + 9801308205744336842784768*b^65 + 925627474287070516239925248*b^64 + 8538722711226571622176522240*b^63 + 424340488354365825007981953024*b^62 + 4635969315442428444656648847360*b^61 + 150178447103547925906703880552448*b^60 + 1606855374708485140986281299279872*b^59 + 30767478333062812211166189321191424*b^58 + 322968502099036093656165169183588352*b^57 - 7416554765387575880271179533755875328*b^56 - 23370120092987155468084874734618017792*b^55 - 9176667210032444410186394578727508901888*b^54 - 66516169335602377902182575813840808706048*b^53 - 3733382629704723534405560522706199316004864*b^52 - 35197962619490025436561170617388908157075456*b^51 - 902882931944290126403920332139666399618400256*b^50 - 8208440501372934659986782980608223505517903872*b^49 - 72972977547525488149147300865245441727532105728*b^48 + 268585229127293356889030925347411332534062022656*b^47 + 81401630081615297830914976514350504696687236218880*b^46 + 758369985869676927075864183688525988706109985128448*b^45 + 50618837693346200950128004093199490534681450758799360*b^44 + 238542028006475340745542437069446706905815282213715968*b^43 + 13697629932519926782938962357717358937743013819822637056*b^42 + 28383610260553398034775016698122228628582612098549284864*b^41 + 764309033332450900774935639842630501010128130111112544256*b^40 - 4675686378783628751620304299148898547821633914225737334784*b^39 - 848562604148705087919805088329626482668575078351736101273600*b^38 - 2182887364042253472075477633843241116794891249870680906924032*b^37 - 303567081135191457276502862973966478912635704121899337982083072*b^36 - 199715567262961125589069797483525359861431769176407323274952704*b^35 - 41142418006169077407879496515683475803884865351137926642329370624*b^34 + 55809774812992776909325294666815776657799760729496929168828039168*b^33 + 4211263433025356733822025697677935532685892401984008264554776464896*b^32 + 16931556361678997225209835390819428502180910907017712933551641969664*b^31 + 2536967020868064217559908146674296301364616324340077771357376308478976*b^30 - 1446828478323813313167958521618162058795394067486733912837740404282112*b^29 + 392772538942063242226014401310973692298145917437787405778383496087909568*b^28 - 1202574045680607476131944926784190356948294611978329871273920977773415488*b^27 + 5013120494126492058741349402135511299467596016559529114670283139148339936*b^26 - 173988476257948851934587970368375052453597305734587514480294800596568115424*b^25 - 7585546335140993852738775547998704088679312692198384187173767279258561902319*b^24 + 492072004738185824399516300600696628592552035477969756740660698644735160704*b^23 - 1037231050565760355032232163720334563670173237708845279988319150242139577528416*b^22 + 3683535222366437945774081509709269863244621425421996860561176292116218669906504*b^21 + 7708784580960898641034703297269833687958871368076561231223549540440688260812026*b^20 + 322802784638763229955857443620959663119895807003010787106110436271876060556414612*b^19 + 15273380821537609288485070407379504282365155447340764719179875825917866869164684474*b^18 - 14878394505291346437482287393934286298120035800404632912584364781546412773385995248*b^17 + 1168530013712645346085083570417995837978319751169031548957738126798980409968469973297*b^16 - 3837143856049175090216012869111483195450003856186458546371730472913093441548949440716*b^15 - 107280610071182333788850393281059166642174013386957712703442914816118814392781525506674*b^14 - 227905195853662781616572977421839584127558158979001852713573777843858570707051706167684*b^13 - 13887576586394867540497089484685831502932365984144605826628655951826679083863536447410687*b^12 + 26549011041275173967947326522624690819770277647072977223259537745513118997115191414589824*b^11 + 496116577676270765594456347857061230726965091602481653505180207979490008773959209199015136*b^10 + 1724429231503359659659802180231062696618872657312060158189655036178083692849601439428199560*b^9 + 60639511718416206110778935602864771126195461662308881106220538996101161141321065504401705304*b^8 - 92639010048243291645547808709132626958524882651439246754724221568163037001651974482559302288*b^7 - 178758010667590109275649446110153021773444458756495505201539674784968829722989296614264632672*b^6 + 133487569805338635546045287907525148729118913309589637711732123816700342155859163883461789152*b^5 + 4648080963563903532466663290151995557816072410468174082490771267384979931613151821995627212560*b^4 + 5969005590570321207248815228465313059942470320281789690321983235435351731514683803613602162048*b^3 + 8063332450905624352036230878019563048044511904397001889567875566799424897238126689393328859520*b^2 + 5937060013940983130007685180257681951781356023613370016727896050048371329840646650336967894912*b + 28291573276910000775851477911409679191220898920721905711289305334194228242581528868936773231936