Number Field Of Degree Research Articles

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.

Local criteria for the unit equation and the asymptotic Fermat’s Last Theorem

Let F be a totally real number field of odd degree. We prove several purely local criteria for the asymptotic Fermat's Last Theorem to hold over F and also, for the nonexistence of solutions to the unit equation over F. For example, if two totally ramifies and three splits completely in F, then the asymptotic Fermat's Last Theorem holds over F.

Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields

In this paper we prove that for every integer [Formula: see text], there exists an explicit constant [Formula: see text] such that the following holds. Let [Formula: see text] be a number field of degree [Formula: see text], let [Formula: see text] be any rational prime that is totally inert in [Formula: see text] and [Formula: see text] any elliptic curve defined over [Formula: see text] such that [Formula: see text] has potentially multiplicative reduction at the prime [Formula: see text] above [Formula: see text]. Then for every rational prime [Formula: see text], [Formula: see text] has an irreducible mod [Formula: see text] Galois representation. This result has Diophantine applications within the “modular method”. We present one such application in the form of an Asymptotic version of Fermat’s Last Theorem that has not been covered in the existing literature.

Unit groups of some multiquadratic number fields of degree 16

Let p and q be two primes and $$\ell$$ be an odd positive square-free integer such that $$\ell \not \in \{1,p, q, pq\}$$ . In this paper we give the unit group of some number fields of the form $${\mathbb {Q}}(\sqrt{2},\sqrt{p}, \sqrt{q}, \sqrt{-\ell } )$$ .

NOTES ON THE K-RATIONAL DISTANCE PROBLEM

AbstractLet K be an algebraic number field. We investigate the K-rational distance problem and prove that there are infinitely many nonisomorphic cubic number fields and a number field of degree n for every $n\geq 2$ in which there is a point in the plane of a unit square at K-rational distances from the four vertices of the square.

A generalization of simplest number fields and their integral basis

An integral basis of the simplest number fields of degree 3,4 and 6 over $\mathbb{Q}$ are well-known, and widely investigated. We generalize the simplest number fields to any degree, and show that an integral basis of these fields is repeating periodically.

Non-vanishing of class group L-functions for number fields with a small regulator

Let $E/\mathbb {Q}$ be a number field of degree $n$. We show that if $\operatorname {Reg}(E)\ll _n |\!\operatorname{Disc}(E)|^{1/4}$ then the fraction of class group characters for which the Hecke $L$-function does not vanish at the central point is $\gg _{n,\varepsilon } |\!\operatorname{Disc}(E)|^{-1/4-\varepsilon }$. The proof is an interplay between almost equidistribution of Eisenstein periods over the toral packet in $\mathbf {PGL}_n(\mathbb {Z})\backslash \mathbf {PGL}_n(\mathbb {R})$ associated to the maximal order of $E$, and the escape of mass of the torus orbit associated to the trivial ideal class.

On the indices in number fields and their computation for small degrees

Let K be a number field. We investigate the indices I(K) and i(K) of K introduced respectively by Dedekind and Gunji-McQuillan. Let n be a positif integer, we then prove that for any prime p ? n, there exists K a number field of degree n over Q such that p divide i(K). This result is an analogue to Bauer''s one for i(K). We compute I(K) and i(K) for cubic fields and infinite families of simplest number fields of degree less than 7. We solve questions and disprove the conjecture stated in [1].

On small discriminants of number fields of degree 8 and 9

We classify all the number fields with signature (4,2), (6,1), (1,4) and (3,3) having discriminant lower than a specific upper bound. This completes the search for minimum discriminants for fields of degree 8 and continues it in the degree 9 case. We recall the theoretical tools and the algorithmic steps upon which our procedure is based, then we focus on the novelties due to a new implementation of this process on the computer algebra system PARI/GP; finally, we make some remarks about the final results, among which the existence of a number field with signature $(3,3)$ and small discriminant which was not previously known.

Units and2-class field towers of some multiquadratic number fields

In this paper, we investigate the unit groups, the $2$-class groups, the $2$-class field towers and the structures of the second $2$-class groups of some multiquadratic number fields of degree $8$ and $16$.

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.

On the error term and zeros of the Dedekind zeta function

Let K be a number field of degree k≥8. In this article, we investigate a certain density theorem for zeros of the Dedekind zeta function attached to K. In this context, our theorem strengthens a result of Heath-Brown. Further, we investigate an upper-bound of the error term for an ideal counting function attached to K. When K is a cyclotomic field, we prove a stronger upper-bound for this error term.

Class field theory, Diophantine analysis and the asymptotic Fermat's Last Theorem

Recent results of Freitas, Kraus, Şengün and Siksek, give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over a specific number field. Those works in turn build on many deep theorems in arithmetic geometry. In this paper we combine the aforementioned results with techniques from class field theory, the theory of p-groups and p-extensions, Diophantine approximation and linear forms in logarithms, to establish the asymptotic Fermat's Last Theorem for many infinite families of number fields, and for thousands of number fields of small degree. For example, we prove the effective asymptotic Fermat's Last Theorem for the infinite family of fields Q(ζ2r)+ where r≥2.

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

The generalized Fermat equation with exponents 2, 3,

We study the generalized Fermat equation $x^{2}+y^{3}=z^{p}$, to be solved in coprime integers, where $p\geqslant 7$ is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve $X(p)$. We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic $p$-torsion modules. Using these criteria we produce the minimal list of twists of $X(p)$ that have to be considered, based on local information at 2 and 3; this list depends on $p\hspace{0.2em}{\rm mod}\hspace{0.2em}24$. We solve the equation completely when $p=11$, which previously was the smallest unresolved $p$. One new ingredient is the use of the ‘Selmer group Chabauty’ method introduced by the third author, applied in an elliptic curve Chabauty context, to determine relevant points on $X_{0}(11)$ defined over certain number fields of degree 12. This result is conditional on the generalized Riemann hypothesis, which is needed to show correctness of the computation of the class groups of five specific number fields of degree 36. We also give some partial results for the case $p=13$. The source code for the various computations is supplied as supplementary material with the online version of this article.

Unramified extensions over low degree number fields

For various nonsolvable groups G, we prove the existence of extensions of the rationals Q with Galois group G and inertia groups of order dividing ge(G), where ge(G) is the smallest exponent of a generating set for G. For these groups G, this gives the existence of number fields of degree ge(G) with an unramified G-extension. The existence of such extensions over Q for all finite groups would imply that, for every finite group G, there exists a quadratic number field admitting an unramified G-extension, as was recently conjectured. We also provide further evidence for the existence of such extensions for all finite groups, by proving their existence when Q is replaced with a function field k(t) where k is an ample field.

Product Formula, Global Relations, and Polyadic Numbers

In the paper, the role of the product formula for nonzero elements of an algebraic number field of finite degree over the field of rationals in several problems of number theory is discussed.

Mean values and moments of arithmetic functions over number fields

For an odd integer $$d > 1$$ and a finite Galois extension $$K/{\mathbb {Q}}$$ of degree d, Lu and Yang (J Number Theory 131:1924–1938, 2011) obtained an asymptotic formula for the mean values of the divisor function for K over square integers. In this article, we obtain the same for finitely many number fields of odd degree and pairwise coprime discriminants, together with the moment of the error term arising here, following the method adapted by Shi (An Stiint Univ Al I Cuza Iasi Mat (N.S.) 62:615–621, 2016). We also define the sum of divisor function over number fields and find the asymptotic behaviour of the summatory function of two number fields taken together.

Integral basis of pure prime degree number fields

Let K = ℚ(θ) be an extension of the field ℚ of rational numbers where θ satisfies an irreducible polynomial xp − a of prime degree belonging to ℤ[x]. In this paper, we give explicilty an integral basis for K using only elementary algebraic number theory. Though an integral basis for such fields is already known (see [Trans. Amer. Math. Soc., 11 (1910), 388–392)], our description of integral basis is different and slightly simpler. We also give a short proof of the formula for discriminant of such fields.

Biquadratic fields having a non-principal Euclidean ideal class

H.W. Lenstra [7] introduced the notion of an Euclidean ideal class, which is a generalization of norm-Euclidean ideals in number fields. Later, families of number fields of small degree were obtained with an Euclidean ideal class (for instance, in [2] and [6]). In this paper, we construct certain new families of biquadratic number fields having a non-principal Euclidean ideal class and this extends the previously known families given by H. Graves [2] and C. Hsu [6].