login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

a(n) is the conductor of extension of Q generated by character values of the n-th alternating group A_n; the smallest m such that the extension is a subfield of the m-th cyclotomic field Q(zeta_m).
3

%I #21 Oct 13 2024 12:21:23

%S 1,1,3,3,5,5,7,105,15,21,231,1155,1365,2145,1155,15015,255255,255255,

%T 285285,4849845,255255,4849845,111546435,111546435,4849845,111546435,

%U 111546435,111546435,3234846615,3234846615,3457939485,100280245065,3234846615,100280245065,100280245065

%N a(n) is the conductor of extension of Q generated by character values of the n-th alternating group A_n; the smallest m such that the extension is a subfield of the m-th cyclotomic field Q(zeta_m).

%C Let g be an element in A_n. The extension of Q generated by chi(g), where chi runs through all irreducible representations of Q_n, is Q unless g has cycle type (lambda_1,...,lambda_k) for distinct odd numbers lambda_1,...,lambda_k, in which case it is Q(sqrt((Product_{i=1..k} lambda_i)*), where m* = (-1)^((m-1)/2)*m.

%C Let Q(G) be the extension of Q generated by character values of a finite group G. For n >= 25, we have Q(A_n) = Q({sqrt((-1)^((p-1)/2)*p) : p odd prime <= n, p != n-2}. This is also true for n <= 5 and for n = 15, 20, 21, 22.

%C The conductor of Q(A_n) (the smallest m such that Q(A_n) is a subfield of the m-th cyclotomic field Q(zeta_m)) is thus Product_{p odd prime <= n, p != n-2} p for n != 6, 7, 10, 11, 14.

%H Jianing Song, <a href="/A376940/b376940.txt">Table of n, a(n) for n = 1..2370</a>

%H Groupprops, <a href="https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_alternating_groups">Linear representation theory of alternating groups</a>.

%H G. R. Robinson and J. G. Thompson, <a href="https://doi.org/10.1006/jabr.1995.1126">Sums of Squares and the Fields Q_{A_n}</a>, Journal of Algebra, vol. 34, issue 1 (May 1995), pp. 225-228.

%H Jianing Song, <a href="/A376938/a376938.txt">The extension Q(A_n) for n <= 24</a>

%e See a-file for Q(A_n) for n <= 24.

%o (PARI) A376940_first_14_terms = [1, 1, 3, 3, 5, 5, 7, 105, 15, 21, 231, 1155, 1365, 2145];

%o a(n) = if(n<=14, A376940_first_14_terms[n], factorback(setminus(primes([3,n]), [n-2])))

%Y Cf. A376938, A376939.

%K nonn,easy

%O 1,3

%A _Jianing Song_, Oct 12 2024