Distribution of Discriminants of Abelian Extensions

Wright, D. J.

doi:10.1112/plms/s3-58.1.17

Cited by 100 publications

(146 citation statements)

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“…The following results give support to the conjecture (see [2], [9], [18], [19], [20], [21], [22], [23], [28], [30]). Theorem 1.3.…”

Section: Conjecture 12supporting

confidence: 60%

“…For the base field K = Q, a complete and explicit solution was given by Mäki in [23]. For a general base field, a solution has been given by Wright in [28], but the problem with his solution is that the constant c K (G), although given as a product of local contributions, cannot be computed explicitly without a considerable amount of additional work. It is always a finite linear combination of Euler products.…”

Section: General Finite Abelian Extensionsmentioning

confidence: 99%

See 1 more Smart Citation

Theory and Experiment of Ultra High Gain Gyrotron Traveling-Wave Amplifier

Chu¹,

Chang²,

Barnett³

et al. 2002

AIP Conference Proceedings

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In this paper we give a survey of recent methods for the asymptotic and exact enumeration of number fields with given Galois group of the Galois closure. In particular, the case of fields of degree up to 4 is now almost completely solved, both in theory and in practice. The same methods also allow construction of the corresponding complete tables of number fields with discriminant up to a given bound.

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“…The following results give support to the conjecture (see [2], [9], [18], [19], [20], [21], [22], [23], [28], [30]). Theorem 1.3.…”

Section: Conjecture 12supporting

confidence: 60%

Section: General Finite Abelian Extensionsmentioning

confidence: 99%

Theory and Experiment of Ultra High Gain Gyrotron Traveling-Wave Amplifier

Chu¹,

Chang²,

Barnett³

et al. 2002

AIP Conference Proceedings

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“…When G is Abelian, this is a theorem of Wright [23]. A deep theorem of Davenport and Heilbronn [5] says that (1.1) holds for the pair (3, S 3 ) with e(3, S 3 ) = 1.…”

Section: ∼ C(d G)x E(dg) Log V(dg) X As X→∞mentioning

confidence: 96%

Densities of quartic fields with even Galois groups

Wong

2005

Proc. Amer. Math. Soc.

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Abstract. Let N (d, G, X) be the number of degree d number fields K with Galois group G and whose discriminant D K satisfies |D K | ≤ X. Under standard conjectures in diophantine geometry, we show that N (4, A 4 , X) X 2/3+ , and that there are N 3+ monic, quartic polynomials with integral coefficients of height ≤ N whose Galois groups are smaller than S 4 , confirming a question of Gallagher. Unconditionally we have N (4, A 4 , X) X 5/6+ , and that the 2-class groups of almost all Abelian cubic fields k have size D 1/3+ k . The proofs depend on counting integral points on elliptic fibrations.

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“…Now, suppose H is abelian; so H = (Z/pZ) r for some prime p and some positive integer r. By [21] we may assume |Q| ≥ 2.…”

Section: Suppose That L/k Is a Galois Extension Withmentioning

confidence: 99%