A376939 - OEIS

A376939

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).

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, 10, 10, 10, 11, 11, 10, 11, 12, 12, 12, 13, 12, 13, 14, 14, 13, 14, 14, 14, 15, 15, 14, 15, 15, 15, 16, 16, 16, 17, 16, 17, 17, 17, 18, 18, 17, 18, 19, 19, 19, 20, 19, 20

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OFFSET

1,8

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

For n >= 25, a(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) = if(n<=24, A376939_first_24_terms[n], primepi(n) - 1 - isprime(n-2))