A343690 - OEIS

%I #11 May 01 2021 02:24:12

%S 1,5,49,117,1609,28037,612233,1257728

%N Minimal value of positive discriminants of number fields of degree n.

%C Conjecture: a(n) < A343772(n) for n == 0, 1 (mod 4), a(n) > A343772(n) for n == 2, 3 (mod 4).

%H LMFDB, <a href="https://www.lmfdb.org/NumberField">Number fields</a>

%H A. M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/doc/zeta.html">Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: A survey of recent results</a>, Sem. Theorie des Nombres, Bordeaux, 2 (1990), pp. 119-141.

%H <a href="/index/Qua#quadfield">Index entries for sequences related to quadratic fields</a>

%F A006557(n) = Min_{a(n), A343772(n)}.

%e The number field F of degree n whose discriminant is positive and of minimal value:

%e n = 2, F = Q[x]/(x^2 - x - 1), d = 5;

%e n = 3, F = Q[x]/(x^3 - x^2 - 2x + 1), d = 49;

%e n = 4, F = Q[x]/(x^4 - x^3 - x^2 + x + 1), d = 117;

%e n = 5, F = Q[x]/(x^5 - x^3 - x^2 + x + 1), d = 1609;

%e n = 6, F = Q[x]/(x^6 - 2x^5 + 3x^3 - 2x - 1), d = 28037;

%e n = 7, F = Q[x]/(x^7 - x^6 + x^5 - x^3 + x^2 - x - 1), d = 612233;

%e n = 8, F = Q[x]/(x^8 - 2x^7 + 4x^5 - 4x^4 + 3x^2 - 2x + 1), d = 1257728.

%Y Cf. A343772 (the negative discriminant case), A006557 (the overall case).

%K nonn,hard,more

%O 1,2

%A _Jianing Song_, Apr 26 2021