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Inspired from this, I tried factor following expressions, Mathematica takes a long time, I couldn't even wait for it to finish. Maple calculating it takes about 5 sec. Can you recommend an efficient method? To get the factored result please click here(generated by Maple 16).

Factor[1 - 216 x^2 - 192 x^3 + 16140 x^4 + 18816 x^5 - 547528 x^6 - 
  687168 x^7 + 8960886 x^8 + 12394752 x^9 - 67518888 x^10 - 
  108989760 x^11 + 178031596 x^12 + 374357376 x^13 + 61149384 x^14 - 
  96214464 x^15 + 3999249 x^16, 
 Extension -> {Sqrt[2], Sqrt[3], Sqrt[5], Sqrt[6], Sqrt[11]}]
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Why dou you think just about this Extension -> {Sqrt[2], Sqrt[3], Sqrt[5], Sqrt[6], Sqrt[11]} ? When you have extended the field of the rationals by Sqrt[2], Sqrt[3] you don't have to extend it as well by Sqrt[6] since there is just Sqrt[6], try e.g. Factor[-6 + x^2, Extension -> {Sqrt[2], Sqrt[3]}]. –  Artes Jan 25 '13 at 12:57
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1 Answer

up vote 2 down vote accepted

Unlike your earlier Expand example, I cannot replicate the slowness here.

In[30]:= Timing[
 fax = Factor[
    1 - 216 x^2 - 192 x^3 + 16140 x^4 + 18816 x^5 - 547528 x^6 - 
     687168 x^7 + 8960886 x^8 + 12394752 x^9 - 67518888 x^10 - 
     108989760 x^11 + 178031596 x^12 + 374357376 x^13 + 
     61149384 x^14 - 96214464 x^15 + 3999249 x^16, 
    Extension -> {Sqrt[2], Sqrt[3], Sqrt[5], Sqrt[6], Sqrt[11]}];]

(* Out[30]= {0.480000, Null} *)

Factor will use a primitive element, so convert to radicals to go to radicals.

In[31]:= Timing[ToRadicals[fax]]

(* Out[31]= {0., 
 157964603341393 + 91200903120336 Sqrt[3] - 
  4278656335470048 Sqrt[5 - 2 Sqrt[6]] + 
  89935414457372160 (5 - 2 Sqrt[6]) + 
  62309081229003360 Sqrt[2] (5 - 2 Sqrt[6]) + 
  51924234357502800 Sqrt[3] (5 - 2 Sqrt[6]) + 
  35974165782948864 Sqrt[6] (5 - 2 Sqrt[6]) - 
  1261240613743683456 (5 - 2 Sqrt[6])^(3/2) - 
  889994874234520384 Sqrt[2] (5 - 2 Sqrt[6])^(3/2) - 
  728177624373698496 Sqrt[3] (5 - 2 Sqrt[6])^(3/2) - 
  513838768562241408 Sqrt[6] (5 - 2 Sqrt[6])^(3/2) + 
  12323061227178959376 (5 - 2 Sqrt[6])^2 + 
  8711905205113320960 Sqrt[2] (5 - 2 Sqrt[6])^2 + 
  7114722584175878784 Sqrt[3] (5 - 2 Sqrt[6])^2 + 
  5029820909052636480 Sqrt[6] (5 - 2 Sqrt[6])^2 - 
  88232583176459376768 (5 - 2 Sqrt[6])^(5/2) - 
  62388546039057394176 Sqrt[2] (5 - 2 Sqrt[6])^(5/2) - 
  50941106398863376896 Sqrt[3] (5 - 2 Sqrt[6])^(5/2) - 
  36020043319228734336 Sqrt[6] (5 - 2 Sqrt[6])^(5/2) + 
  478093358287174151680 (5 - 2 Sqrt[6])^3 + 
  338062333338701947008 Sqrt[2] (5 - 2 Sqrt[6])^3 + 
  276027326042542835520 Sqrt[3] (5 - 2 Sqrt[6])^3 + 
  195180381321361818624 Sqrt[6] (5 - 2 Sqrt[6])^3 - 
  1997428926525420492288 (5 - 2 Sqrt[6])^(7/2) - 
  1412395246615804708608 Sqrt[2] (5 - 2 Sqrt[6])^(7/2) - 
  1153216137412904493312 Sqrt[3] (5 - 2 Sqrt[6])^(7/2) - 
  815446769474433867264 Sqrt[6] (5 - 2 Sqrt[6])^(7/2) + 
  6493163212229734802016 (5 - 2 Sqrt[6])^4 + 
  4591359616509582336000 Sqrt[2] (5 - 2 Sqrt[6])^4 + 
  3748829510010630067200 Sqrt[3] (5 - 2 Sqrt[6])^4 + 
  2650822723593059823360 Sqrt[6] (5 - 2 Sqrt[6])^4 - 
  16446760524212038838784 (5 - 2 Sqrt[6])^(9/2) - 
  11629615898963258318848 Sqrt[2] (5 - 2 Sqrt[6])^(9/2) - 
  9495541639814490759168 Sqrt[3] (5 - 2 Sqrt[6])^(9/2) - 
  6714361852964445470208 Sqrt[6] (5 - 2 Sqrt[6])^(9/2) + 
  32268730179416053862400 (5 - 2 Sqrt[6])^5 + 
  22817437911549382464000 Sqrt[2] (5 - 2 Sqrt[6])^5 + 
  18630360044435013600000 Sqrt[3] (5 - 2 Sqrt[6])^5 + 
  13173653928269969006592 Sqrt[6] (5 - 2 Sqrt[6])^5 - 
  48352419038274050377728 (5 - 2 Sqrt[6])^(11/2) - 
  34190323359888173177856 Sqrt[2] (5 - 2 Sqrt[6])^(11/2) - 
  27916282124786355016704 Sqrt[3] (5 - 2 Sqrt[6])^(11/2) - 
  19739792411726278232064 Sqrt[6] (5 - 2 Sqrt[6])^(11/2) + 
  53964237338898693931264 (5 - 2 Sqrt[6])^6 + 
  38158478229689721446400 Sqrt[2] (5 - 2 Sqrt[6])^6 + 
  31156267008017112299520 Sqrt[3] (5 - 2 Sqrt[6])^6 + 
  22030807639741785308160 Sqrt[6] (5 - 2 Sqrt[6])^6 - 
  43021093037533864040448 (5 - 2 Sqrt[6])^(13/2) - 
  30420506556750775984128 Sqrt[2] (5 - 2 Sqrt[6])^(13/2) - 
  24838239593682148958208 Sqrt[3] (5 - 2 Sqrt[6])^(13/2) - 
  17563287686455991494656 Sqrt[6] (5 - 2 Sqrt[6])^(13/2) + 
  22800236715739400232960 (5 - 2 Sqrt[6])^7 + 
  16122202025212411435008 Sqrt[2] (5 - 2 Sqrt[6])^7 + 
  13163722830612297722880 Sqrt[3] (5 - 2 Sqrt[6])^7 + 
  9308157661371798208512 Sqrt[6] (5 - 2 Sqrt[6])^7 - 
  7001994355390356627456 (5 - 2 Sqrt[6])^(15/2) - 
  4951157685773279219712 Sqrt[2] (5 - 2 Sqrt[6])^(15/2) - 
  4042603322067924430848 Sqrt[3] (5 - 2 Sqrt[6])^(15/2) - 
  2858552225425724940288 Sqrt[6] (5 - 2 Sqrt[6])^(15/2) + 
  888100660971628040448 (5 - 2 Sqrt[6])^8 + 
  627981998862511964160 Sqrt[2] (5 - 2 Sqrt[6])^8 + 
  512745154955400278016 Sqrt[3] (5 - 2 Sqrt[6])^8 + 
  362565576601493852160 Sqrt[6] (5 - 2 Sqrt[6])^8 - 
  2470283462936832 Sqrt[2 (5 - 2 Sqrt[6])] - 
  2470283462936832 Sqrt[3 (5 - 2 Sqrt[6])] - 
  1426218778490016 Sqrt[6 (5 - 2 Sqrt[6])]} *)

At this point one might attempt denesting. There was a post on this topic in Mathematica.StackExchange maybe a couple of months ago.

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Here: mathematica.stackexchange.com/a/15926/5 –  rm -rf Jan 25 '13 at 15:53
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