Prime Ideals of a Galois Number Field and its Subfields

1998 ◽  
pp. 79-88
Author(s):  
David Hilbert
Keyword(s):  
Number Field ◽  
Prime Ideals ◽  
Galois Number

scholarly journals A Density of Ramified Primes

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Stephanie Chan ◽  
Christine McMeekin ◽  
Djordjo Milovic
Keyword(s):  
Number Field ◽  
Class Number ◽  
Joint Distribution ◽  
Number Fields ◽  
Character Sums ◽  
Prime Ideals ◽  
Odd Degree ◽  
Narrow Class

AbstractLet K be a cyclic number field of odd degree over $${\mathbb {Q}}$$ Q with odd narrow class number, such that 2 is inert in $$K/{\mathbb {Q}}$$ K / Q . We define a family of number fields $$\{K(p)\}_p$$ { K ( p ) } p , depending on K and indexed by the rational primes p that split completely in $$K/{\mathbb {Q}}$$ K / Q , in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in $$K(p)/{\mathbb {Q}}$$ K ( p ) / Q is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension $$K/{\mathbb {Q}}$$ K / Q . Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.

Prime splitting in abelian number fields and linear combinations of Dirichlet characters

2014 ◽  
Vol 10 (04) ◽  
pp. 885-903 ◽  
Author(s):  
Paul Pollack
Keyword(s):  
Number Field ◽  
Nonzero Element ◽  
Number Fields ◽  
Upper Bounds ◽  
Prime Ideals ◽  
Linear Combinations ◽  
Dirichlet Characters ◽  
Abelian Number Field ◽  
Splitting Type ◽  
A Finite Group

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].

ON A THEOREM OF DEDEKIND

2008 ◽  
Vol 04 (06) ◽  
pp. 1019-1025 ◽  
Author(s):  
SUDESH K. KHANDUJA ◽  
MUNISH KUMAR
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Minimal Polynomial ◽  
Algebraic Number Field ◽  
Rational Numbers ◽  
Prime Ideals ◽  
Algebraic Integers ◽  
Dedekind Domains ◽  
Modulo P ◽  
Polynomial Formula

Let K = ℚ(θ) be an algebraic number field with θ in the ring AK of algebraic integers of K and f(x) be the minimal polynomial of θ over the field ℚ of rational numbers. For a rational prime p, let [Formula: see text] be the factorization of the polynomial [Formula: see text] obtained by replacing each coefficient of f(x) modulo p into product of powers of distinct monic irreducible polynomials over ℤ/pℤ. Dedekind proved that if p does not divide [AK : ℤ[θ]], then the factorization of pAK as a product of powers of distinct prime ideals is given by [Formula: see text], with 𝔭i = pAK + gi(θ)AK, and residual degree [Formula: see text]. In this paper, we prove that if the factorization of a rational prime p in AK satisfies the above-mentioned three properties, then p does not divide [AK:ℤ[θ]]. Indeed the analogue of the converse is proved for general Dedekind domains. The method of proof leads to a generalization of one more result of Dedekind which characterizes all rational primes p dividing the index of K.

scholarly journals On Dedekind's criterion and monogenicity over Dedekind rings

2003 ◽  
Vol 2003 (71) ◽  
pp. 4455-4464 ◽  
Author(s):  
M. E. Charkani ◽  
O. Lahlou
Keyword(s):  
Number Field ◽  
Valuation Ring ◽  
Minimal Polynomial ◽  
Discrete Valuation Ring ◽  
Integral Closure ◽  
Prime Ideals ◽  
Dedekind Ring ◽  
Ring Of Integers ◽  
Integral Element

We give a practical criterion characterizing the monogenicity of the integral closure of a Dedekind ringR, based on results on the resultantRes(p,pi)of the minimal polynomialpof a primitive integral element and of its irreducible factorspimodulo prime ideals ofR. We obtain a generalization and an improvement of the Dedekind criterion (Cohen, 1996), and we give some applications in the case whereRis a discrete valuation ring or the ring of integers of a number field, generalizing some well-known classical results.

scholarly journals Average Frobenius distribution for elliptic curves defined over finite Galois extensions of the rationals

2011 ◽  
Vol 150 (3) ◽  
pp. 439-458 ◽  
Author(s):  
KEVIN JAMES ◽  
ETHAN SMITH
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Fixed Number ◽  
Prime Ideals ◽  
Galois Extensions ◽  
Trace Of Frobenius

AbstractLet K be a fixed number field, assumed to be Galois over ℚ. Let r and f be fixed integers with f positive. Given an elliptic curve E, defined over K, we consider the problem of counting the number of degree f prime ideals of K with trace of Frobenius equal to r. Except in the case f = 2, we show that ‘on average,’ the number of such prime ideals with norm less than or equal to x satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang–Trotter conjecture and extends the work of several authors.

A note on prime divisors of polynomials P(Tk); k ≥ 1

2019 ◽  
Vol 69 (1) ◽  
pp. 213-222
Author(s):  
François Legrand
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Residue Field ◽  
Sufficient Conditions ◽  
Integral Closure ◽  
Prime Ideals ◽  
Prime Divisors ◽  
Reduction Modulo

Abstract Let F be a number field, OF the integral closure of ℤ in F, and P(T) ∈ OF[T] a monic separable polynomial such that P(0) ≠ 0 and P(1) ≠ 0. We give precise sufficient conditions on a given positive integer k for the following condition to hold: there exist infinitely many non-zero prime ideals 𝓟 of OF such that the reduction modulo 𝓟 of P(T) has a root in the residue field OF/𝓟, but the reduction modulo 𝓟 of P(Tk) has no root in OF/𝓟. This makes a result from a previous paper (motivated by a problem in field arithmetic) asserting that there exist (infinitely many) such integers k more precise.

scholarly journals Average Frobenius distribution for the degree two primes of a number field

2013 ◽  
Vol 154 (3) ◽  
pp. 499-525 ◽  
Author(s):  
KEVIN JAMES ◽  
ETHAN SMITH
Keyword(s):  
Elliptic Curve ◽  
Number Field ◽  
Prime Ideals ◽  
Trace Of Frobenius

AbstractLet K be a number field and r an integer. Given an elliptic curve E, defined over K, we consider the problem of counting the number of degree two prime ideals of K with trace of Frobenius equal to r. Under certain restrictions on K, we show that “on average” the number of such prime ideals with norm less than or equal to x satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang–Trotter conjecture and extends the work of several authors.

A note on the least prime that splits completely in a nonabelian Galois number field

2018 ◽  
Vol 292 (1-2) ◽  
pp. 183-192
Author(s):  
Zhenchao Ge ◽  
Micah B. Milinovich ◽  
Paul Pollack
Keyword(s):  
Number Field ◽  
Galois Number
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