A376940 - OEIS

%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