A368056 Degrees of number fields unramified away from 2.

1, 2, 4, 8, 16, 17

Offset: 1

Author

Robin Visser, Dec 09 2023

Keywords

• nonn
• hard
• more

Comments

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.

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A367643, A367669, A368057.