Prime splitting in abelian number fields and linear combinations of Dirichlet characters
2014 ◽
Vol 10
(04)
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pp. 885-903
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Keyword(s):
Let 𝕏 be a finite group of primitive Dirichlet characters. Let ξ = ∑χ∈𝕏 aχ χ be a nonzero element of the group ring ℤ[𝕏]. We investigate the smallest prime q that is coprime to the conductor of each χ ∈ 𝕏 and that satisfies ∑χ∈𝕏 aχ χ(q) ≠ 0. Our main result is a nontrivial upper bound on q valid for certain special forms ξ. From this, we deduce upper bounds on the smallest unramified prime with a given splitting type in an abelian number field. For example, let K/ℚ be an abelian number field of degree n and conductor f. Let g be a proper divisor of n. If there is any unramified rational prime q that splits into g distinct prime ideals in ØK, then the least such q satisfies [Formula: see text].
2018 ◽
Vol 14
(09)
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pp. 2333-2342
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Keyword(s):
1987 ◽
Vol 107
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pp. 135-146
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Keyword(s):
2006 ◽
Vol 05
(01)
◽
pp. 35-41
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1960 ◽
Vol 17
◽
pp. 171-179
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Keyword(s):
Keyword(s):
2016 ◽
Vol 0
(0)
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