On Newton polygons techniques and factorization of polynomials over Henselian fields

A class of soluble groups

These is a survey on the theory of height zeta functions, written on the occasion of a French-Japanese winter school, held in Miura (Kanagawa, Japan) in Jan. 2008. It does not presuppose much knowledge in algebraic geometry. The last chapter of the survey explains recent results obtained in collaboration with Yuri Tschinkel concerning asymptotics of volumes of height balls in analytic geometry over local fields, or in adelic spaces.

Let $${f(x)=(x-a_1)\cdots (x-a_m)}$$ , where a 1, . . . , a m are distinct rational integers. In 1908 Schur raised the question whether f(x) ± 1 is irreducible over the rationals. One year later he asked whether $${(f(x))^{2^k}+1}$$ is irreducible for every k ≥ 1. In 1919 Polya proved that if $${P(x)\in\mathbb{Z}[x]}$$ is of degree m and there are m rational integer values a for which 0 < |P(a)| < 2−N N! where $${N=\lceil m/2\rceil}$$ , then P(x) is irreducible. A great number of authors have published results of Schur-type or Polya-type afterwards. Our paper contains various extensions, generalizations and improvements of results from the literature. To indicate some of them, in Theorem 3.1 a Polya-type result is established when the ground ring is the ring of integers of an arbitrary imaginary quadratic number field. In Theorem 4.1 we describe the form of the factors of polynomials of the shape h(x) f(x) + c, where h(x) is a polynomial and c is a constant such that |c| is small with respect to the degree of h(x) f(x). We obtain irreducibility results for polynomials of the form g(f(x)) where g(x) is a monic irreducible polynomial of degree ≤ 3 or of CM-type. Besides elementary arguments we apply methods and results from algebraic number theory, interpolation theory and diophantine approximation.

Let E be a local field, i.e., a field which is complete with respect to a rank one discrete valuation υ (we do not require any finiteness condition on the residue class field of E). Let f(X) be a polynomial in one variable, with coefficients in E. It is well known [4, 6, 9, 11, 13] that the Newton polygon method allows us to gather information about the factorization of f(X). This method consists of attaching to each side S of a Newton polygon of f(X) a factor (not necessarily irreducible) of f(X), the degree of which is the length of the horizontal projection of S.

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Heigh Ho! Says Tall-y A frog he would a-wooing go Hoorah! For David O. Tall. David Tall died on 15th July, 2024 at the age of 83. He was one of the most influential members of the mathematics education community as it was establishing itself and growing through the 1980s and 90s. In particular, he was critical to establishing research in undergraduate mathematics education as a distinct subfield within a discipline previously focused on schools. Despite extensive international travel for academic work, David was a man of the Midlands in the UK, having been born and schooled in Northamptonshire, studying at Oxford and, after a short period as a lecturer at the University of Sussex, moving to Kenilworth in 1969 for a post at the then new Warwick University where he spent the rest of his long and distinguished academic career. In his early years at Warwick's Mathematics Institute, he formed a writing partnership with Ian Stewart, producing highly influential textbooks such as The Foundations of Mathematics (1977), Algebraic Number Theory (1979) (later editions incorporating new material to become Algebraic Number Theory and Fermat's Last Theorem) (2001) and Complex Analysis (1983). These are still set texts in many UK universities many decades later. David had completed a PhD on the representation of groups at Oxford, supervised by Michael Atiyah (later a Fields and Abel prize winner), but David already had a deep interest in how people think about mathematics and learn it. After moving from the Warwick Mathematics Institute to the Mathematics Education Research Centre at Warwick, he took the decision to undertake a second PhD with Richard Skemp as

In his biography of Emil Artin, Richard Brauer describes the years from 1931–1941 as a time when “Artin spoke through his students and through the members of his mathematical circle” rather than through written publications. This paper explores these seemingly quiet years when Artin immigrated to America and disseminated ideas about algebraic number theory during this time in his collaboration with George Whaples, a young American mathematician who had just completed his Ph.D. at the University of Wisconsin. The main result of their work is the use of the product formula for valuations to come up with an axiomatic characterization of both algebraic number fields and algebraic function fields with a finite field of constants. These two families of fields are exactly the fields for which class field theory is known to hold. We situate their mathematical work in the broader context of algebraic number theory and their lives within the broader historical context.

To put the issues in a broader context, these three number-theoretic problems were instrumental in the emergence of algebraic number theory-one of the two main sources of the modern discipline of commutative algebra.' The other source was algebraic geometry. It was in the setting of these two subjects that many of the main concepts and results of commutative algebra evolved in the 19th century. Thus commutative algebra can be said to have been well developed before it was created. To set the scene for our story, a word about mathematics, and especially algebra, in the 19th century. The period witnessed! fundamental transformations in mathematics -in its concepts, its methods, and in mathematicians' attitude toward their subject. These included a growing insistence on rigor and abstraction; a predisposition for founding general theories rather than focusing on specific problems; an acceptance of nonconstructive existence proofs; the emergence of mathematical specialties and specialists; the rise of the view of mathematics as a free human activity, neither deriving from, nor dependent on, nor necessarily applicable to concrete setting; the rise of set-theoretic thinking; and the reemergence of (a new variant of) the axiomatic method. As for algebra, its subject-matter and methods changed beyond recognition. Algebra as the study of solvability of polynomial equations gave way to algebra as the study of abstract structures defined axiomatically. While in the past algebra was on the periphery of mathematics, it now (the 19th and early 20th centuries) became one of its central concerns. Moreover, algebra began to penetrate other mathematical fields (e.g. geometry, analysis, logic, topology, number theory) to such an extent that in the early decades of the 20th century one began to speak of the algebraization of mathematics ([3, p. 135] and [22]).

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Let D be an integral domain with identity and let K denote the quotient field of D. If P is a prime ideal of D denote by $P[[X]]$ that prime ideal of $D[[X]]$ consisting of all those formal power series each of whose coefficients belongs to P. In this paper the following question is considered: When is ${(D[[X]])_{P[[X]]}}$ a valuation ring? Our main theorem states that if ${(D[[X]])_{P[[X]]}}$ is a valuation ring, then ${D_P}$ must be a rank one discrete valuation ring. Moreover, we show that if ${D_P}$ is a rank one discrete valuation ring and if $PD[[X]] = P[[X]]$, then ${(D[[X]])_{P[[X]]}}$ is a valuation ring. We also give an example to show that ${(D[[X]])_{P[[X]]}}$ need not be a valuation ring when ${D_P}$ is rank one discrete.

The positive integer points and integral points of elliptic curves are very important in the theory of number and arithmetic algebra, it has a wide range of applications in cryptography and other fields. There are some results of positive integer points of elliptic curve y2 = x3 + ax + b, a, b ∈ Z In 1987, D. Zagier submit the question of the integer points on y2 = x3 − 27x + 62, it count a great deal to the study of the arithmetic properties of elliptic curves.In 2009, Zhu H L and Chen J H solved the problem of the integer points on y2 = x3 − 27x + 62 by using algebraic number theory and P-adic analysis method.In 2010, By using the elementary method, Wu H M obtain all the integral points of elliptic curves y2 = x3 − 27x − 62.In 2015, Li Y Z and Cui B J solved the problem of the integer points on y2 = x3 − 21x − 90 By using the elementary method.In 2016, Guo J solved the problem of the integer points on y2 = x3 + 27x + 62 by using the elementary method.In 2017, Guo J proved that y2 = x3 − 21x + 90 has no integer points by using the elementary method.Up to now, there is no relevant conclusions on the integral points of elliptic curves y2 = x3 + 3x − 14, which is the subject of this paper.By using congruence and Legendre Symbol, it can be proved that elliptic curve y2 = x3 + 3x − 14 has only one integer point: (x, y) = (2, 0).

The geometry. of numbers can be traced back at least to Lagrange [1773], who proved important results about quadratic forms in two variables. The proofs as well as the formulations of results were purely arithmetic. Reviewing a book of Seeber [1831B] on ternary quadratic forms, Gauß [1831] introduced for the first time geometric methods. Geometric methods were predominant in the work of Dirichlet [1850]. On the other hand Hermite [1850] and Korkine and Zolotareff [1872], [1873], [1877] gave arithmetic proofs for their results on quadratic forms in more than three variables. Finally Minkowski [1891] noticed that a simple geometric argument which he used to give a new proof of a theorem of Hermite could be adapted to much more general situations. Then Minkowski [1896B], [7B], [11B] started a systematic study of geometric methods in number theory and called this new branch of number theory geometry of numbers. Many results and most concepts of modern geometry of numbers have their origin in the work of Minkowski. After Minkowski many eminent mathematicians made contributions to this field. In order to avoid controversies I will not mention any of them. Geometry of numbers is closely related to other branches of number theory such as algebraic number theory and Diophantine approximation. A flourishing offspring is discrete geometry, developed mainly by Fejes Tóth and his school.

This book is a translation into English of Hilbert's Theorie der algebraischen Zahlkorper best known as the Zahlbericht, first published in 1897, in which he provided an elegantly integrated overview of the development of algebraic number theory up to the end of the nineteenth century. The Zahlbericht provided also a firm foundation for further research in the subject. It is based on the work of the great number theorists of the nineteenth century. The Zahlbericht can be seen as the starting point of all twentieth century investigations in algebraic number theory, reciprocity laws and class field theory. For this English edition an Introduction has been added by F. Lemmermeyer and N. Schappacher.

PurposeThe authors have determined whether the points fixed by all the full and the partial Atkin–Lehner involutions WQ on X0(N) for N ≤ 50 are Weierstrass points or not.Design/methodology/approachThe design is by using Lawittes's and Schoeneberg's theorems.FindingsFinding all Weierstrass points on X0(N) fixed by some Atkin–Lehner involutions. Besides, the authors have listed them in a table.Originality/valueThe Weierstrass points have played an important role in algebra. For example, in algebraic number theory, they have been used by Schwartz and Hurwitz to determine the group structure of the automorphism groups of compact Riemann surfaces of genus g ≥ 2. Whereas in algebraic geometric coding theory, if one knows a Weierstrass nongap sequence of a Weierstrass point, then one is able to estimate parameters of codes in a concrete way. Finally, the set of Weierstrass points is useful in studying arithmetic and geometric properties of X0(N).

A global approach to phase calibration is presented. The corresponding theoretical framework calls on elementary concepts of algebraic graph theory (spanning tree of maximal weight, cycles) and algebraic number theory (lattice, nearest lattice point). The traditional approach can thereby be better understood. In radio imaging and in optical interferometry, the self-calibration procedures must often be conducted with much care. The analysis presented should then help in finding a better compromise between the coverage of the calibration graph (which must be as complete as possible) and the quality of the solution (which must of course be reliable).