## Counting cyclic quartic extensions of a number field.(English)Zbl 1090.11068

Let $$G$$ be a finite group and $$K$$ some number field. Let $$N_K(G,X)$$ denote the number of all Galois extensions $$L/K$$ with Galois group $$G$$ such that the norm of the relative discriminant is $$\leq X$$. By a conjecture of Malle it is expected that there are constants $$a_K, b_K, c_K$$ (depending on $$G$$) such that $N_K(G,X) = c_K X^{a_K} (\log X)^{b_K-1}.$ The constants $$a_K$$ and $$b_K$$ were determined by D. J. Wright [Proc. Lond. Math. Soc. (3) 58, No. 1, 17–50 (1989; Zbl 0628.12006)] for all abelian groups; in this article, the authors determine $$c_K$$ in the case where $$G$$ is the cyclic group of order $$4$$.

### MSC:

 11R16 Cubic and quartic extensions 11R45 Density theorems 11R29 Class numbers, class groups, discriminants 11R32 Galois theory

### Keywords:

density; discriminants; cyclic quartic extensions; ramification

Zbl 0628.12006
Full Text:

### References:

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