A376938 - OEIS

A376938

a(n) is the degree of extension of Q generated by character values of the n-th alternating group A_n.

1, 1, 2, 2, 2, 2, 2, 4, 2, 2, 4, 8, 8, 8, 16, 16, 32, 32, 32, 128, 64, 128, 128, 128, 128, 256, 256, 256, 512, 512, 512, 1024, 512, 1024, 1024, 1024, 2048, 2048, 1024, 2048, 4096, 4096, 4096, 8192, 4096, 8192, 16384, 16384, 8192, 16384, 16384, 16384, 32768, 32768, 16384

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OFFSET

1,3

COMMENTS

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.

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.

LINKS

Jianing Song, Table of n, a(n) for n = 1..10000

Groupprops, Linear representation theory of alternating groups.

G. R. Robinson and J. G. Thompson, Sums of Squares and the Fields Q_{A_n}, Journal of Algebra, vol. 34, issue 1 (May 1995), pp. 225-228.

Jianing Song, The extension Q(A_n) for n <= 24

FORMULA

a(n) = 2^A376939(n); for n >= 25, A376939(n) is the number of odd primes p <= n, p != n-2.

EXAMPLE

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

PROG

(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];

a(n) = 2^if(n<=24, A376939_first_24_terms[n], primepi(n) - 1 - isprime(n-2))