0,1,2

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a(n) is the number of quadratic number fields Q[(sqrt(d)] ) (including Q itself) that are subfields of the cyclotomic field Q[(exp(Pi*i/n)], ), where i is the imaginary unit. Note that for odd k, Q[(exp(2*Pi*i/k)] ) = Q[(exp(2*Pi*i/(2*k))], ), so we can just consider the case Q[(exp(2*Pi*i/(2*k))] ) for integers k and let n = 2*k.

List of quadratic number fields (including Q itself) that are subfields of Q[(exp(Pi*i/n)]):

n = 2 (the quotient field over the Gaussian integers): Q, Q[(i]);

n = 3 (the quotient field over the Eisenstein integers): Q, Q[(sqrt(-3)]);

n = 4: Q, Q[(sqrt(2)], ), Q[(i], ), Q[(sqrt(-2)]);

n = 5: Q, Q[(sqrt(5)]);

n = 6: Q, Q[(sqrt(3)], ), Q[(sqrt(-3)], ), Q[(i]);

n = 7: Q, Q[(sqrt(-7)]);

n = 8: Q, Q[(sqrt(2)], ), Q[(i], ), Q[(sqrt(-2)]);

n = 9: Q, Q[(sqrt(-3)]);

n = 10: Q, Q[(sqrt(5)], ), Q[(i], ), Q[(sqrt(-5)]);

n = 11: Q, Q[(sqrt(-11)]);

n = 12: Q, Q[(sqrt(2)], ), Q[(sqrt(3)], ), Q[(sqrt(6)], ), Q[(sqrt(-3)], ), Q[(i], ), Q[(sqrt(-2)], ), Q[(sqrt(-6)]);

n = 13: Q, Q[(sqrt(13)]);

n = 14: Q, Q[(sqrt(7)], ), Q[(i], ), Q[(sqrt(-7)]);

n = 15: Q, Q[(sqrt(5)], ), Q[(sqrt(-3)], ), Q[(sqrt(-15)]);

n = 16: Q, Q[(sqrt(2)], ), Q[(i], ), Q[(sqrt(-2)]).

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1, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 4, 4, 4, 2, 4, 2, 8, 4, 4, 2, 8, 2, 4, 2, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 8, 2, 8, 4, 4, 2, 8, 2, 4, 4, 8, 2, 4, 4, 8, 4, 4, 2, 16, 2, 4, 4, 4, 4, 8, 2, 8, 4, 8, 2, 8, 2, 4, 4, 8, 4, 8, 2, 8, 2, 4, 2, 16, 4, 4, 4, 8, 2, 8, 4, 8, 4, 4, 4, 8, 2, 4, 4, 8

a(n) is the number of quadratic number fields Q[sqrt(d)] (including Q itself) that are subfields of the cyclotomic field Q[exp(Pi*i/n)], where i is the imaginary unit. Note that for odd k, Q[exp(2*Pi*i/k)] = Q[exp(2*Pi*i/(2*k))], so we can just consider the case Q[exp(2*Pi*i/(2*k))] for integers k and let n = 2*k.

a(n) = 2 if and only if n = 2 or n = p^e, where p is an odd prime and e >= 1.

Multiplicative with a(2) = 2 and a(2^e) = 4 for e > 1; a(p^e) = 2 for odd primes p.

a(n) = 2^omega(n) if 4 does not divide n, otherwise 2^(omega(n)+1), omega = A001221.

List of quadratic number fields (including Q itself) that are subfields of Q[exp(Pi*i/n)]:

n = 2 (the quotient field over the Gaussian integers): Q, Q[i];

n = 3 (the quotient field over the Eisenstein integers): Q, Q[sqrt(-3)];

n = 4: Q, Q[sqrt(2)], Q[i], Q[sqrt(-2)];

n = 5: Q, Q[sqrt(5)];

n = 6: Q, Q[sqrt(3)], Q[sqrt(-3)], Q[i];

n = 7: Q, Q[sqrt(-7)];

n = 8: Q, Q[sqrt(2)], Q[i], Q[sqrt(-2)];

n = 9: Q, Q[sqrt(-3)];

n = 10: Q, Q[sqrt(5)], Q[i], Q[sqrt(-5)];

n = 11: Q, Q[sqrt(-11)];

n = 12: Q, Q[sqrt(2)], Q[sqrt(3)], Q[sqrt(6)], Q[sqrt(-3)], Q[i], Q[sqrt(-2)], Q[sqrt(-6)];

n = 13: Q, Q[sqrt(13)];

n = 14: Q, Q[sqrt(7)], Q[i], Q[sqrt(-7)];

n = 15: Q, Q[sqrt(5)], Q[sqrt(-3)], Q[sqrt(-15)];

n = 16: Q, Q[sqrt(2)], Q[i], Q[sqrt(-2)].

(PARI) a(n) = 2^#znstar(2*n)[2]

Cf. A060594, A001221.

allocated for Jianing Song

a(n) = A060594(2n).

1, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 8

0,2

a(n) = 2*A060594(n) if n is even and not divisible by 8, otherwise A060594(n).

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nonn,mult

Jianing Song, Sep 23 2019

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allocated for Jianing Song

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