Comparing the density of $D_4$ and $S_4$ quartic extensions of number fields
| Autor: | Matthew Friedrichsen, Daniel Keliher |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
11R42
11R29 11R45 11R16 Mathematics - Number Theory Mathematics::Number Theory Applied Mathematics General Mathematics Galois group Algebraic number field Upper and lower bounds Combinatorics Mathematics::Algebraic Geometry Quadratic equation Discriminant Mathematics::Quantum Algebra Quartic function FOS: Mathematics Number Theory (math.NT) Mathematics::Representation Theory Mathematics |
| Zdroj: | Proceedings of the American Mathematical Society. 149:2357-2369 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/proc/15358 |
| Popis: | When ordered by discriminant, it is known that about 83% of quartic fields over Q have associated Galois group S_4, while the remaining 17% have Galois group D_4. We study these proportions over a general number field F. We find that asymptotically 100% of quadratic number fields have more D_4 extensions than S_4 and that the ratio between the number of D_4 and S_4 quartic extensions is biased arbitrarily in favor of D_4 extensions. Under GRH, we give a lower bound that holds for general number fields. Fixed a typo with Theorem 1.3 that is present in the published version of the paper. The main results remain unchanged |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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