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%I #20 Oct 13 2024 12:21:28
%S 0,0,1,1,1,1,1,2,1,1,2,3,3,3,4,4,5,5,5,7,6,7,7,7,7,8,8,8,9,9,9,10,9,
%T 10,10,10,11,11,10,11,12,12,12,13,12,13,14,14,13,14,14,14,15,15,14,15,
%U 15,15,16,16,16,17,16,17,17,17,18,18,17,18,19,19,19,20,19,20
%N 2^a(n) is the degree of extension of Q generated by character values of the n-th alternating group A_n; a(n) = log_2 A376938(n).
%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.
%H Jianing Song, <a href="/A376939/b376939.txt">Table of n, a(n) for n = 1..10000</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>
%F For n >= 25, a(n) is the number of odd primes p <= n, p != n-2.
%e See a-file for Q(A_n) for n <= 24.
%o (PARI) A376939_first_24_terms = [0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 3, 3, 4, 4, 5, 5, 5, 7, 6, 7, 7, 7];
%o a(n) = if(n<=24, A376939_first_24_terms[n], primepi(n) - 1 - isprime(n-2))
%Y Cf. A376938, A376940.
%K nonn,easy
%O 1,8
%A _Jianing Song_, Oct 12 2024