Cube root Ramanujan formulas and elementary Galois theory

The cube root Ramanujan formulas are explained from the point of view of Galois theory. Let F be a cyclic cubic extension of a field K. It is proved that the normal closure over K of a pure cubic extension of F contains a certain pure cubic extension of K. The proposed proof can be generalized to radicals of any prime degree q. In the case where the base field K is the field of rational numbers and the field F is embedded in the cyclotomic extension obtained by adding the pth roots of unity, the corresponding simple radical extension of the field of rational numbers is explicitly constructed. The proof of the main result illustrates Hilbert’s Theorem 90. An example of a particular formula generalizing Ramanujan’s formulas for degree 5 is given. A necessary condition for nested radical expressions of depth 2 to be contained in the normal closure of a pure cubic extension of the field F is given.     

Construction of genus field and some applications

Let k be a number field of finite degree. The narrow genus field K of k (genus field of k in the sense of Fröhlich) is defined as the maximal extension of k which is unramified at all finite primes of k of the form kk₁, where k₁ is an Abelian number field. In this article, K is determined and some applications are given. The results indicate a possibility that many class field theoretic properties of normal number fields could be extended to nonnormal number fields.     

K3 surfaces, rational curves, and rational points

We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981.     

Some results on the capitulation problem

Let K be an unramified abelian extension of a number field F with Galois group G. K corresponds to a subgroup H of the ideal class group of F. We study the subgroup J of ideal classes in H which become trivial in K. There is an epimorphism from the cohomology group H^?1(G, Cl_K) to J which is an isomorphism if G is cyclic; Cl_K is the ideal class group of K. Some results on the structure of J and Cl_K are obtained.     

Fiercely ramified cyclic extensions of p-adic fields with imperfect residue field

We study the ramification of fierce cyclic Galois extensions of a local field K of characteristic zero with a one-dimensional residue field of characteristic p > 0. Using Kato’s theory of the refined Swan conductor, we associate to such an extension a ramification datum, consisting of a sequence of pairs (δ _i , ω _i ), where δ _i is a positive rational number and ω _i a differential form on the residue field of K. Our main result gives necessary and sufficient conditions on such sequences to occur as a ramification datum of a fierce cyclic extension of K.     

Sur l'indépendance l-adique de nombres algébriques

Let l a prime number and K a Galois extension over the field of rational numbers, with Galois group G. A conjecture is put forward on l-adic independence of algebraic numbers, which generalizes the classical ones of Leopoldt and Gross, and asserts that the l-adic rank of a G submodule of K^x depends only on the character of its Galois representation. When G is abelian and in some other cases, a proof is given of this conjecture by using l-adic transcendence results.     

All double coverings of cyclotomic fields

A double covering of a Galois extension K/F in the sense of [3] is an extension /K of degree ≤2 such that /F is Galois. In this paper we determine explicitly all double coverings of any cyclotomic extension over the rational number field in the complex number field. We get the results mainly by Galois theory and by using and modifying the results and the methods in [2] and [3]. Project 10571097 supported by NSFC     

First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers

We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidabilitv. We also obtain a structural sufficient condition for definability of the ring of integers over its field of fractions. In particular, we show that the following propositions hold: (1) For any rational prime q and any positive rational integer m. algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by q^m. (2) Given a prime q, and an integer m > 0, algebraic integers are definable in a cyclotomic extension (and any of its subfields) generated by any set \(\{ {\zeta _{{p^l}}}|l \in {Z_{ > 0,}}P \ne q\) is any prime such that q^{m +1} ∧(p — 1)}. (3) The first-order theory of Any Abelina Extension of Q With Finitely Many Rational Primes is undecidable and rational integers are definable in these extensions.We also show that under a condition on the splitting of one rational Q generated elliptic curve over the field in question is enough to have a definition of Z and to show that the field is undecidable.     

Some applications of the Hales-Jewett theorem to field arithmetic

Let K be a field whose absolute Galois group is finitely generated. If K neither finite nor of characteristic 2, then every hyperelliptic curve over K with all of its Weierstrass points defined over K has infinitely many K-points. If, in addition, K is not an algebraic extension of a finite field, then every elliptic curve over K with all of its 2-torsion rational has infinite rank over K. These and similar results are deduced from the Hales-Jewett theorem.     

The trace-form of an algebraic number field

Let F be the rational field or a p-adic field, and let K an algebraic number field over F. If ω₁,…, ω_n is an integral basis for the ring $\text{D}$_L of integers in K, then the quadratic form Q whose matrix is (trace_K∥F(ω_iω_j)) has integral coefficients, and is called an integral trace-form. Q is determined by K up to integral equivalence. The purpose of this paper is to show that the genus of Q determines the ramification of primes in K.     

On a question of Frey and Jarden about the rank of Abelian varieties

This paper deals with a classical question of Frey and Jarden, who asked in their 1974 paper if any non-zero Abelian variety over a number field K acquires infinite rank over the maximal Abelian extension Kab of the ground field. We generalize recent results of Rosen and Wong on the subject. However, the original question in full generality remains open. Some further results on the rank in certain other infinite extensions are included.     

Elementary Abelianp-extensions of algebraic function fields

LetK be a field of characteristicp>0 andF/K be an algebraic function field. We obtain several results on Galois extensionsE/F with an elementary Abelian Galois group of orderp ⁿ.

1. E can be generated overF by some elementy whose minimal polynomial has the specific formT ^pn?T?z.
2. A formula for the genus ofE is given.
3. IfK is finite, then the genus ofE grows much faster than the number of rational points (as [E∶F] → ∞).
4. We present a new example of a function fieldE/K whose gap numbers are nonclassical.

     

On the existence ofp-units and minkowski units in totally real cyclic fields

LetK be a totally real cyclic number field of degree n > 1. A unit inK is called an m-unit, if the index of the group generated by its conjugations in the group U*_K of all units modulo ±1 is coprime tom. It is proved thatK contains an m-unit for every m coprime to n. The mutual relationship between the existence of m-units and the existence of a Minkowski unit is investigated for those n for which the class number hФ(ζ_n) of the n-th cyclotomic field is equal to 1. For n which is a product of two distinct primes p and q, we derive a sufficient condition for the existence of a Minkowski unit in the case when the field K contains a p-unit for every prime p, namely that every ideal contained in a finite list (see Lemma 11) is principal. This reduces the question of whether the existence of a p-unit and a q-unit implies the existence of a Minkowski unit to a verification of whether the above ideals are principal. As a corollary of this, we establish that every totally real cyclic field K of degree n = 2q, where q = 2, 3 or 5, contains a Minkowski unit if and only if it contains a 2-unit and a q-unit.     

Places of algebraic function fields in arbitrary characteristic

We consider the Zariski space of all places of an algebraic function field F|K of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime divisors, places of maximal rank, zero-dimensional discrete places) lie dense in this topology. Further, we give several equivalent characterizations of fields that are large, in the sense of F. Pop's Annals paper Embedding problems over large fields. We also study the question whether a field K is existentially closed in an extension field L if L admits a K-rational place. In the appendix, we prove the fact that the Zariski space with the Zariski topology is quasi-compact and that it is a spectral space.     

G-Closed fields and imbeddings of quadratic number fields

A field, K, that has no extensions with Galois group isomorphic to G is called G-closed. It is proved that a finite extension of K admits an infinite number of nonisomorphic extensions with Galois group G. A trinomial of degree n is exhibited with Galois group, the symmetric group of degree n, and with prescribed discriminant. This result is used to show that any quadratic extension of an A_n-closed field admits an extension with Galois group A_n.     

An embedding theorem for Eichler orders

Let B be a quaternion algebra over number field K. Assume that B satisfies the Eichler condition (i.e., there is at least one archimedean place which is unramified in B). Let Ω be an order in a quadratic extension L of K. The Eichler orders of B which admit an embedding of Ω are determined. This is a generalization of Chinburg and Friedman's embedding theorem for maximal orders.     

The Schur group of an abelian number field

We characterize the maximum r-local index of a Schur algebra over an abelian number field K in terms of global information determined by the field K for an arbitrary rational prime, r. This completes and unifies previous results of Janusz in [G.J. Janusz, The Schur group of an algebraic number field, Ann. of Math. (2) 103 (1976) 253-281] and Pendergrass in [J.W. Pendergrass, The 2-part of the Schur group, J. Algebra 41 (1976) 422-438].     

Regular subsets of valued fields and Bhargava’s v-orderings

We generalize notions and results obtained by Amice for regular compact subsets S of a local field K and extended by Bhargava to general compact subsets of K. Considering any ultrametric valued field K and subsets S that are regular in a generalized sense (but not necessarily compact), we show that they still have strong properties such as having v-orderings ${\{a_n\}_{n\geq0}}$ which satisfy a generalized Legendre formula, which are very well ordered and well distributed sequences in the sense of Helsmoortel and which remain v-orderings when a finite number of the initial terms of the sequence are deleted.     

Function field genus theory for non-Kummer extensions

In this paper we first obtain the genus field of a finite abelian non-Kummer l–extension of a global rational function field. Then, using that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields and our previous results, we deduce the general expression of the genus field of a finite abelian extension of a global rational function field.