For D4-Galois extensions of Q, the number with all four Gundlach multi-invariants bounded by X1, X2, X3, X4 is asymptotic to (27/8) times the Euler product over odd primes of (1-1/p)^4(1+4/p) times X1X2X3X4, provided the X_i are not too far apart in logarithmic size.
Counting biquadratic number fields with quaternionic and dihedral extensions
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
abstract
We establish asymptotic formulae for the number of biquadratic number fields of bounded discriminant that can be embedded into a quaternionic or a dihedral extension. To prove these results, we express the solvability of these inverse Galois problems in terms of Hilbert symbols, and then apply a method of Heath-Brown to bound sums of linked quadratic characters.
fields
math.NT 1years
2025 1verdicts
CONDITIONAL 1representative citing papers
citing papers explorer
-
Counting $D_4$-field extensions by multi-invariants
For D4-Galois extensions of Q, the number with all four Gundlach multi-invariants bounded by X1, X2, X3, X4 is asymptotic to (27/8) times the Euler product over odd primes of (1-1/p)^4(1+4/p) times X1X2X3X4, provided the X_i are not too far apart in logarithmic size.