Number Field Of Prime Degree Research Articles

The unit equation over cyclic number fields of prime degree

Let $\ell \ne 3$ be a prime. We show that there are only finitely many cyclic number fields $F$ of degree $\ell$ for which the unit equation $$\lambda + \mu = 1, \qquad \lambda,~\mu \in \mathcal{O}_F^\times$$ has solutions. Our result is effective. For example, we deduce that the only cyclic quintic number field for which the unit equation has solutions is $\mathbb{Q}(\zeta_{11})^+$.

Torsion of elliptic curves with rational 𝑗-invariant defined over number fields of prime degree

Let [ K : Q ] = p [K:\mathbb {Q}]=p be a prime number and let E / K E/K be an elliptic curve with j ( E ) ∈ Q j(E) \in \mathbb {Q} . We determine the all possibilities for E ( K ) t o r s E(K)_{tors} . We obtain these results by studying Galois representations of E E and of its quadratic twists.

The Trace Form Over Cyclic Number Fields

AbstractIn the mid 80’s Conner and Perlis showed that for cyclic number fields of prime degree p the isometry class of integral trace is completely determined by the discriminant. Here we generalize their result to tame cyclic number fields of arbitrary degree. Furthermore, for such fields, we give an explicit description of a Gram matrix of the integral trace in terms of the discriminant of the field.

Genus numbers of cyclic and dihedral extensions of prime degree

We study genus numbers of cyclic and dihedral number fields of prime degree $l\geq 5$. For cyclic number fields, we obtain definitive results. For the dihedral case, by assuming a conjecture on the average of $l$-part class numbers, we obtain partial resu

A mild Tchebotarev theorem for [formula omitted

It is well known that the Tchebotarev density theorem implies that an irreducible ℓ-adic representation ρ of the absolute Galois group of a number field K is determined (up to isomorphism) by the characteristic polynomials of Frobenius elements at any set of primes of density 1. In this Note we make some progress on the automorphic side for GL(n) by showing that, for any cyclic extension K/k of number fields of prime degree p, a cuspidal automorphic representation π of GL(n,AK) is determined up to twist equivalence, even up to isomorphism if p=2, by the knowledge of its local components at the (density one) set SK/k of primes of K of degree 1 over k. The proof uses the Luo–Rudnick–Sarnak bound, certain L-functions of positive type, Kummer theory, and automorphic descent along suitable nested sequences of cyclic p2-extensions.

The smallest inert prime in a cyclic number field of prime degree

Fix an odd prime `. For each cyclic extension K/Q of degree `, let nK denote the least rational prime which is inert in K, and let rK be the least rational prime which splits completely in K. We show that nK possesses a finite mean value, where the average is taken over all such K ordered by conductor. As an example (` = 3), the average least inert prime in a cyclic cubic field is approximately 2.870. We conjecture that rK also has a finite mean value, and we prove this assuming the Generalized Riemann Hypothesis. For the case ` = 3, we give an unconditional proof that the average of rK exists and is about 6.862.

NORM-EUCLIDEAN CYCLIC FIELDS OF PRIME DEGREE

Let K be a cyclic number field of prime degree ℓ. Heilbronn showed that for a given ℓ there are only finitely many such fields that are norm-Euclidean. In the case of ℓ = 2 all such norm-Euclidean fields have been identified, but for ℓ ≠ 2, little else is known. We give the first upper bounds on the discriminants of such fields when ℓ > 2. Our methods lead to a simple algorithm which allows one to generate a list of candidate norm-Euclidean fields up to a given discriminant, and we provide some computational results.

Decomposition of places in dihedral and cyclic quintic trinomial extensions of global fields

In this paper, we give a complete and explicit description of the splitting behavior of any place in a quintic trinomial dihedral or cyclic extension of a rational function field of finite characteristic distinct from 2 and 5. Our characterization depends only on the order of the base field and a parametrization of the coefficients of the generating trinomial. Moreover, we contrast some of our results to trinomial dihedral number fields of prime degree, where the unit rank behaves quite differently from the function field scenario.

On correspondence between orders and ideals with a normal basis in cyclic subfields of Q(ξp) of a prime degree l

Abstract Let K be a tamely ramified cyclic algebraic number field of prime degree l. In the paper one-to-one correspondence between all orders of K with a normal basis and all ideals of K with a normal basis is given.

Finding Normal Integral Bases of Cyclic Number Fields of Prime Degree

Let L be a cyclic number field of prime degree p. In this paper we study how to compute efficiently a normal integral basis for L, if there is at least one, assuming that an integral basisΓ for L is known. We reduce our problem to the problem of finding the generator of a principal ideal in thep th cyclotomic field.

A hyper-Kloosterman sum identity

From a Davenport-Hasse identity of Gauss sums an identity of a hyper-Kloosterman sum has been deduced. Using this identity the theory of Kloosterman sheaves and equidistribution of hyper-Kloosterman sums can be applied to an exponential sum over a cyclic algebraic number field of prime degree. This identity might also be applied to base change problems in representation theory via a possible relative trace formula over the cyclic number field.

Power Integral Bases in Composits of Number Fields

AbstractIn the present paper we consider the problem of finding power integral bases in number fields which are composits of two subfields with coprime discriminants. Especially, we consider imaginary quadratic extensions of totally real cyclic number fields of prime degree. As an example we solve the index form equation completely in a two parametric family of fields of degree 10 of this type.

Small generators of number fields

If K is a number field of degree n over Q with discriminant DK and if α∈K generates K, i.e. K=Q(α), then the height of α satisfies \(\) with \(\). The paper deals with the existence of small generators of number fields in this sense. We show: (1) For each $n$ there are infinitely many number fields K of degree $n$ with a generator α such that \(\). (2) There is a constant d2 such that every imaginary quadratic number field has a generator α which satisfies \(\).¶(3) If K is a totally real number field of prime degree n then one can find an integral generator α with \(\).

The lifting of an exponential sum to a cyclic algebraic number field of prime degree

The lifting of an exponential sum to a cyclic algebraic number field of prime degree

Solvability of norm equations over cyclic number fields of prime degree

Let L = Q[α] be an abelian number field of prime degree q, and let α be a nonzero rational number. We describe an algorithm which takes as input α and the minimal polynomial of α over Q, and determines if α is a norm of an element of L. We show that, if we ignore the time needed to obtain a complete factorization of α and a complete factorization of the discriminant of α, then the algorithm runs in time polynomial in the size of the input. As an application, we give an algorithm to test if a cyclic algebra A = (E, σ, σ) over Q is a division algebra.

A remark about zeta functions of number fields of prime degree.

A remark about zeta functions of number fields of prime degree.

On the group of units of an absolutely cyclic number field of prime degree

On the group of units of an absolutely cyclic number field of prime degree

On the class number of an absolutely cyclic number field of prime degree

On the class number of an absolutely cyclic number field of prime degree