Densities of Quartic Fields with Even Galois Groups
نویسندگان
چکیده
Let N(d,G,X) be the number of degree d number fields K with Galois group G and whose discriminant DK satisfies |DK | ≤ X. Under standard conjectures in diophantine geometry, we show that N(4, A4, X) X2/3+ , and that there are N3+ monic, quartic polynomials with integral coefficients of height ≤ N whose Galois groups are smaller than S4, confirming a question of Gallagher. Unconditionally we haveN(4, A4, X) X5/6+ , and that the 2-class groups of almost all Abelian cubic fields k have size D k . The proofs depend on counting integral points on elliptic fibrations.
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