CoCalc Shared Filesclass group - extension of degree 3.sagewsOpen in CoCalc with one click!
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K.<a,b> = NumberField([x^2-2,x^2-3]) OK = K.ring_of_integers() OK.basis()
[(-3/2*b + 17/2)*a - 7*b + 4, 6*a - 5*b, -2*b*a + 5, 11*a - 9*b]
P = x^3-x^2-2*x-8 K.<a> = NumberField(P) OK = K.ring_of_integers() OK.basis()
[1, 1/2*a^2 + 1/2*a, a^2]
R.<x> = QQ[] # 4,6,7, 10 #P = x^3+10*x+1 #P = x^3+4*x+1 P = x^2+29 #L = P.splitting_field('b'); #L.galois_group(type="pari").group() #exit(0); K.<a> = NumberField(P) ['is galois : ',K.is_galois()] OK = K.ring_of_integers() C = OK.class_group() C I = C.gen() I I.ideal().basis() OK.basis() (1+a).norm() Ga = K.galois_group(names='y') ga = Ga.gen(0) T.<z> = K[] p = z^3+10*z+1 L.<b> = p.splitting_field(); Gb = L.galois_group(names='y') gb = Gb.gen(0) #U.<t> = ZZ[] (7+a).norm() #OK.norm #I = OK.ideal(3,a^2+a+1); I #for m in [1..11] : # P = x^3+m*x+1 # isP = P.is_prime() # ['is prime : ', isP] # if isP : # K.<a> = NumberField(P) # #['is galois : ',K.is_galois()] # OK = K.ring_of_integers() # OK.class_group() #OK.class_group(0).gen() #I = OK.ideal(3,a^2+a+1); I #I.basis()
['is galois : ', True] Class group of order 6 with structure C6 of Number Field in a with defining polynomial x^2 + 29 Fractional ideal class (3, a + 2) [3, a + 2] [1, a] 30 78
OK.basis() for k in [1..6] : I^k ga type(ga) ga(ga(a)) K.galois_group(type="pari").group()
[1, a] Fractional ideal class (3, a + 2) Fractional ideal class (5, a + 4) Fractional ideal class (2, a + 1) Fractional ideal class (5, a + 1) Fractional ideal class (3, a + 1) Trivial principal fractional ideal class (1,2) <class 'sage.rings.number_field.galois_group.GaloisGroupElement'> a PARI group [2, -1, 1, "S2"] of degree 2