A396866 Order of the Pólya group of real quadratic field with discriminant A003658(n), n >= 2.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2

Offset

2,12

Comments

The Pólya group Po(K) of a number field K is the subgroup of class group Cl(K) generated by {products of prime ideals of O_K with norm q : q prime powers}. For K being a quadratic field, it is clear that Po(K) is generated by prime ideals lying above each ramified prime, and Po(K) is an elementary abelian 2-group. This sequence gives orders of the Pólya groups of real quadratic fields.

Links

Jianing Song, Table of n, a(n) for n = 2..10000

Jean-Luc Chabert, From Pólya fields to Pólya groups (I) Galois extensions, Journal of Number Theory, 2019, 203, pp.360-375.

Formula

If K is the real quadratic field with discriminant A003658(n), then a(n) = 2^(A317991(n) - 1) if the fundamental unit of K has norm 1, 2^A317991(n) otherwise. (See Proposition 1.4 in the Chabert link). Note that A317991(n) = omega(D) - 1.

Prog

(PARI) Po_2_rank(D) = omega(D) - 1 - (norm(quadunit(D))==1) \\ gives 2-rank of Po(D) for fundamental D
for(D=1, 1000, if(D>1 && isfundamental(D), print1(2^Po_2_rank(D), ", ")))

Crossrefs

Cf. A003658, A317989, A391426, A396865, A396868 (earliest occurrences of each number).

Cf. A319659 (for imaginary quadratic fields).

Sequence in context: A336137 A371735 A088323 * A391426 A003652 A071625

Adjacent sequences: A396863 A396864 A396865 * A396867 A396868 A396869

Keyword

nonn

Author

Jianing Song, Jun 08 2026

Status

approved