A368057 -id:A368057 - cp's OEIS frontend

A367669 Number of degree 3 number fields unramified outside the first n prime numbers.

0, 9, 32, 108, 360, 1168, 3638, 11492, 35638, 111059

Offset: 1

Robin Visser, Nov 26 2023

B. Matschke showed that a(11) = 340618 assuming the Generalized Riemann Hypothesis.

			For n = 1, there are no cubic number fields unramified away from 2, so a(1) = 0.
For n = 2, the a(2) = 9 cubic number fields unramified away from {2,3} can be given by Q(a) where a is a root of x^3 - 3x - 1, x^3 - 2, x^3 + 3x - 2, x^3 - 3, x^3 - 3x - 4, x^3 - 3x - 10, x^3 - 12, x^3 - 6, or x^3 - 9x - 6.

K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), no. 219, 1213-1237.
J. W. Jones and D. P. Roberts, A database of number fields, LMS J. Comput. Math. 17 (2014), no. 1, 595-618.
B. Matschke, Number fields unramified outside S.

Cf. A126646 (degree 2), A368057 (degree 4).

A368056 Degrees of number fields unramified away from 2.

Original entry on oeis.org

1, 2, 4, 8, 16, 17

Offset: 1

Robin Visser, Dec 09 2023

Every power of 2 appears in this sequence, as for any positive integer n, adjoining a primitive 2^(n+1)-th root of unity to Q yields a degree 2^n number field unramified away from 2.

The first example of an odd degree number field unramified away from 2 is the degree 17 number field Q(a) where a is a root of the polynomial x^17 - 2x^16 + 8x^13 + 16x^12 - 16x^11 + 64x^9 - 32x^8 - 80x^7 + 32x^6 + 40x^5 + 80x^4 + 16x^3 - 128x^2 - 2x + 68, found by David Harbater.

			For n = 1, a(1) = 1 as the unique degree 1 number field (the rationals) is unramified everywhere.
For n = 2, a(2) = 2 as there exists a degree 2 number field unramified away from 2 (for example Q(i), Q(sqrt(2)), or Q(sqrt(-2))).
For n = 3, a(3) = 4 as there exists a degree 4 number field unramified away from 2 (for example, adjoining a fourth root of 2 to Q), but there does not exist any such degree 3 number field.

D. Harbater, Galois groups with prescribed ramification, In Arithmetic geometry (Tempe, AZ, 1993) (Vol. 174, pp. 35-60). Amer. Math. Soc., Providence, RI.
J. Jones, Number fields unramified away from 2, J. Number Theory 130 (2010), no. 6, 1282-1291.
J. R. Merriman and N. P. Smart, Curves of genus 2 with good reduction away from 2 with a rational Weierstrass point, Math. Proc. Cambridge Philos. Soc. 114 (1993), no. 2, 203-214.

Cf. A367643, A367669, A368057.

cp's OEIS Frontend

A367669 Number of degree 3 number fields unramified outside the first n prime numbers.

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