A341443 - OEIS

%I #7 Feb 12 2021 16:35:00

%S 1,2,2,5,3,5,4,8,4,8,4,16,6,10,8,11,5,10,6,27,10,10,4,27,6,16,6,32,6,

%T 27,8,14,10,14,16,32,9,15,16,54,8,32,8,32,16,10,4,38,8,16,14,54,6,15,

%U 16,54,15,16,4,118,12,20,30,17,30,32,8,49,10,54,8,54,12,24,16,48,20,54,8,81

%N a(n) is the number of subfields of the (2n)-th cyclotomic field Q(zeta_(2n)) = Q(exp(2*Pi*i/(2n))).

%C This is another version of A272831: here we only consider cyclotomic fields of even order since Q(zeta_n) = Q(zeta_(2n)) for odd n.

%C a(n) is the number of subgroups of the multiplicative group of integers modulo 2n.

%H ResearchGate, <a href="https://www.researchgate.net/post/How_many_subfields_does_a_cyclotomic_field_have">How many subfields does a cyclotomic field have</a>

%F a(n) = A272831(2n).

%e a(1) = 1: Q(zeta_2) = Q, which has itself as the only subfield.

%e a(2) = 2: Q(zeta_4) = Q(i) has 2 subfields: Q and Q(i).

%e a(3) = 2: Q(zeta_6) = Q(sqrt(-3)) has 2 subfields: Q and Q(sqrt(-3)).

%e a(4) = 5: Q(zeta_8) = Q(i, sqrt(2)) has 5 subfields: Q, Q(i), Q(sqrt(2)), Q(sqrt(-2)) and Q(zeta_8).

%e a(5) = 3: Q(zeta_10) = Q(zeta_5) has 3 subfields: Q, Q(sqrt(5)), Q(zeta_5).

%e a(6) = 5: Q(zeta_12) = Q(i, sqrt(-3)) has 5 subfields: Q, Q(i), Q(sqrt(-3)), Q(sqrt(3)) and Q(zeta_12).

%e a(7) = 4: Q(zeta_14) = Q(zeta_7) has 4 subfields: Q, Q(sqrt(-7)), Q((zeta_7)+(zeta_7)^(-1)) and Q(zeta_7).

%o (GAP) List([1..80], n->Sum( ConjugacyClassesSubgroups( LatticeSubgroups( GL(1, ZmodnZ(2*n)))), Size));

%Y Cf. A272831.

%K nonn

%O 1,2

%A _Jianing Song_, Feb 12 2021