Package: pari-stable
Version: 2.7.4

In the example below, nfsubfields says that the field defined by f12 is primitive, but nfisincl confirms that the quartic field defined by f4 is a subfield.

John

GP/PARI CALCULATOR Version 2.7.4 (released)

amd64 running linux (x86-64/GMP-5.0.2 kernel) 64-bit version

compiled: Jun 24 2015, gcc version 4.8.4 (Ubuntu 4.8.4-2ubuntu1~14.04)

threading engine: single

(readline v6.3 enabled, extended help enabled)

PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY

WARRANTY WHATSOEVER.

Type ? for help, \q to quit.

Type ?12 for how to get moral (and possibly technical) support.

parisize = 128000000, primelimit = 500000

? f4=x^4 + 4*x^2 + 169

%1 = x^4 + 4*x^2 + 169

? f12=x^12 - 4*x^11 - 14*x^10 + 44*x^9 + 226*x^8 + 380*x^7 - 542*x^6 - 2620*x^5 + 4249*x^4 - 6312*x^3 + 13056*x^2 - 67392*x + 97344

%2 = x^12 - 4*x^11 - 14*x^10 + 44*x^9 + 226*x^8 + 380*x^7 - 542*x^6 - 2620*x^5 + 4249*x^4 - 6312*x^3 + 13056*x^2 - 67392*x + 97344

? nfsubfields(f12)

%3 = [[x, 0], [x^12 - 4*x^11 - 14*x^10 + 44*x^9 + 226*x^8 + 380*x^7 - 542*x^6 - 2620*x^5 + 4249*x^4 - 6312*x^3 + 13056*x^2 - 67392*x + 97344, x]]

? nfisincl(f4,f12)

%4 = [-313345065/2512202169568*x^11 + 1086348149/3768303254352*x^10 + 9139699651/3768303254352*x^9 - 1415549251/628050542392*x^8 - 128077187921/3768303254352*x^7 - 184263212867/1884151627176*x^6 - 1401016439/25634716016*x^5 + 583328371861/1884151627176*x^4 - 1130295363727/7536606508704*x^3 + 12379626061/179443012112*x^2 - 1067669250/1914788239*x + 63417240637/12077895046, -180234539/3768303254352*x^11 + 306142255/3768303254352*x^10 + 222530500/235518953397*x^9 - 220127029/942075813588*x^8 - 6164509997/471037906794*x^7 - 10264383601/235518953397*x^6 - 6280695611/134582259084*x^5 + 36528039997/942075813588*x^4 - 737509158053/3768303254352*x^3 - 131407676923/179443012112*x^2 - 4540732071/7659152956*x + 43285952459/12077895046, 180234539/3768303254352*x^11 - 306142255/3768303254352*x^10 - 222530500/235518953397*x^9 + 220127029/942075813588*x^8 + 6164509997/471037906794*x^7 + 10264383601/235518953397*x^6 + 6280695611/134582259084*x^5 - 36528039997/942075813588*x^4 + 737509158053/3768303254352*x^3 + 131407676923/179443012112*x^2 + 4540732071/7659152956*x - 43285952459/12077895046, 313345065/2512202169568*x^11 - 1086348149/3768303254352*x^10 - 9139699651/3768303254352*x^9 + 1415549251/628050542392*x^8 + 128077187921/3768303254352*x^7 + 184263212867/1884151627176*x^6 + 1401016439/25634716016*x^5 - 583328371861/1884151627176*x^4 + 1130295363727/7536606508704*x^3 - 12379626061/179443012112*x^2 + 1067669250/1914788239*x - 63417240637/12077895046]