OFFSET
2,1
COMMENTS
The old definition was "Minimal discriminant of number field of degree n."
From Jianing Song, Apr 29 2021: (Start)
Minimal absolute value of discriminants of number fields with signature r_1 = n - 2, r_2 = 1. 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.
This is the second column of A343290, negated. (End)
REFERENCES
H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 617.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
LMFDB, Number fields
A. M. Odlyzko, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: A survey of recent results, Sem. Theorie des Nombres, Bordeaux, 2 (1990), pp. 119-141.
EXAMPLE
From Jianing Song, Apr 29 2021: (Start)
The number field F of degree n with exactly 2 complex embeddings (signature r_1 = n - 2, r_2 = 1) whose discriminant is of minimal absolute value:
n = 2, F = Q[x]/(x^2 - x + 1), d = -3;
n = 3, F = Q[x]/(x^3 - x^2 + 1), d = -23;
n = 4, F = Q[x]/(x^4 - x^3 + 2x - 1), d = -275;
n = 5, F = Q[x]/(x^5 - x^3 - 2x^2 + 1), d = -4511;
n = 6, F = Q[x]/(x^6 - x^5 - 2x^4 + 3x^3 - x^2 - 2x + 1), d = -92779;
n = 7, F = Q[x]/(x^7 - 3x^5 - x^4 + x^3 + 3x^2 + x - 1), d = -2306599. (End)
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
EXTENSIONS
Definition clarified by Jianing Song, Apr 29 2021
STATUS
approved