Demo 6 Ramification groups of global fields.sagews

#sage only has the ability to compute ramification groups for global fields.  However any extension of local fields can be expressed as a completion of an extension of global fields, and the ramification groups are the same in both cases.
K.<zeta16>= CyclotomicField(2^4); K #construct the extension Q(zeta_16) of Q.  The ramification groups we get here will be the same as the ramification groups of the local extension Q_2(zeta_16)/Q_2

Cyclotomic Field of order 16 and degree 8

G = K.galois_group(); G #this is not super informative

Galois group of Cyclotomic Field of order 16 and degree 8

G.list() #lists the elements of G based on how they permute the Galois conjugates of zeta_16.  

[(), (1,2,3,4)(5,6,7,8), (1,3)(2,4)(5,7)(6,8), (1,4,3,2)(5,8,7,6), (1,5)(2,6)(3,7)(4,8), (1,6,3,8)(2,7,4,5), (1,7)(2,8)(3,5)(4,6), (1,8,3,6)(2,5,4,7)]

[g(zeta16) for g in G.list()] #see how the elements of G act on zeta16.  unfortunately this is not in a nice order.

[zeta16, zeta16^5, -zeta16, -zeta16^5, -zeta16^7, -zeta16^3, zeta16^7, zeta16^3]

p = K.primes_above(2)[0]; p #choose the unique prime above 2, we'll calculate the ramification group

Fractional ideal (2, zeta16 + 1)

p.decomposition_group() #p is the only prime above 2, so the decompsition group is the whole Galois group

Galois group of Cyclotomic Field of order 16 and degree 8

p.inertia_group() #2 is totally ramified in K, so the inertia group is also everything

Galois group of Cyclotomic Field of order 16 and degree 8

p.ramification_group(0) #the ramification group G_0 is equal to the inertia group by definition

Galois group of Cyclotomic Field of order 16 and degree 8

p.ramification_group(1) #G1 is the tame inertia group, which is the 2-sylow subgroup of G_0.  Since G_0 is already a 2-group, G_1 = G_0

Galois group of Cyclotomic Field of order 16 and degree 8

G2 = p.ramification_group(2); G2 #now we have an order 4 subgroup

Subgroup [(), (1,2,3,4)(5,6,7,8), (1,3)(2,4)(5,7)(6,8), (1,4,3,2)(5,8,7,6)] of Galois group of Cyclotomic Field of order 16 and degree 8

G2.fixed_field() #the fixed field is Q(i), though for some reason Sage writes it as Q(4*i)

(Number Field in zeta160 with defining polynomial x^2 + 16, Ring morphism: From: Number Field in zeta160 with defining polynomial x^2 + 16 To: Cyclotomic Field of order 16 and degree 8 Defn: zeta160 |--> 4*zeta16^4)

[g(zeta16) for g in G2] #the orbit of zeta_16 under G2 is {zeta_16^i : i is 1 mod 4}

[zeta16, zeta16^5, -zeta16, -zeta16^5]

G3 = p.ramification_group(3); G3 #G_3  = G_2

Subgroup [(), (1,2,3,4)(5,6,7,8), (1,3)(2,4)(5,7)(6,8), (1,4,3,2)(5,8,7,6)] of Galois group of Cyclotomic Field of order 16 and degree 8

G4 = p.ramification_group(4); G4 #but G_4 has order 2

Subgroup [(), (1,3)(2,4)(5,7)(6,8)] of Galois group of Cyclotomic Field of order 16 and degree 8

G4.fixed_field() #the fixed field is Q(zeta_8), which sage writes as Q(2*zeta8)

(Number Field in zeta160 with defining polynomial x^4 + 16, Ring morphism: From: Number Field in zeta160 with defining polynomial x^4 + 16 To: Cyclotomic Field of order 16 and degree 8 Defn: zeta160 |--> 2*zeta16^2)

[g(zeta16) for g in G4] #and the orbit of zeta_16 under G4 is {zeta_16^i : i is 1 mod 8}

[zeta16, -zeta16]