A343290 - OEIS

%I #28 May 01 2021 10:00:18

%S 1,5,-3,49,-23,725,-275,117,14641,-4511,1609,300125,-92779,28037,

%T -9747,20134393,-2306599,612233,-184607,282300416

%N Irregular triangle read by rows: For n >= 1, 0 <= k <= floor(n/2), T(n,k) is the minimal discriminant (in absolute value) of the number fields with signature r_1 = n - 2*k, r_2 = k.

%C For a number field F with degree n, the signature of F is a pair of numbers (r_1, r_2), where r_1 is the number of real embeddings of F, r_2 is half the number of complex embeddings of F. Obviously, we have r_1 + 2*r_2 = n. For example, a real quadratic field has r_1 = 2, r_2 = 0, and an imaginary quadratic field has r_1 = 0, r_2 = 1.

%C T(0,4) = 1257728, T(9,0) = 9685993193.

%C The sign of T(n,k) is (-1)^k.

%C It seems that the terms of each row are strictly decreasing in absolute value.

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

%F A006557(n) = Min_{k=1..floor(n/2)} |T(n,k)|.

%F A343690(n) = Min_{k=1..floor(n/2), k even} |T(n,k)|.

%F A343772(n) = Min_{k=1..floor(n/2), k odd} |T(n,k)|, n >= 2.

%e Let F be a field with signature r_1 = 5, r_2 = 0, then disc(F) >= 14641. The equality holds when F = Q[x]/(x^5 - x^4 - 4x^3 + 3x^2 + 3^x - 1), so T(5,0) = 14641.

%e Let F be a field with signature r_1 = 3, r_2 = 1, then disc(F) <= -4511. The equality holds when F = Q[x]/(x^5 - x^3 - 2x^2 + 1), so T(5,1) = -4511.

%e Let F be a field with signature r_1 = 7, r_2 = 0, then disc(F) >= 20134393. The equality holds when F = Q[x]/(x7 - x^6 - 6x^5 + 4x^4 + 10x^3 - 4x^2 - 4x + 1), so T(7,0) = 20134393.

%Y Cf. A006557, A343690, A343772.

%Y First column is A006554. Second column is A006555 (negated).

%K sign,hard,more

%O 1,2

%A _Jianing Song_, Apr 28 2021