Root Modulo Research Articles

Polynomials with roots modulo every integer

Given a polynomial with integer coefficients, we calculate the density of the set of primes modulo which the polynomial has a root. We also give a simple criterion to decide whether or not the polynomial has a root modulo every non-zero integer.

On class numbers of algebraic function fields defined by y2 = xn + a over GF( p)

In this study, the class number for a hyperelliptic function field of genus g, constant field K = GF( p) and equation y 2 = x n + a (a ∃ K ∗) has been obtained as h = p g + 1, where n and are prime numbers greater than 2 and p is congruent to a primitive root modulo n.

The modern algebraic method

The suggestion for this talk comes from the Deutsche Mathematiker-Vereinigung. Its aim is to promote modem algebra among non-specialists. Those who are familiar with it will not learn anything new from my talk; rather, I will try to give expression to their convictions. My objective is not to induce anybody to leave his own field of specialization and to become an algebraist; I see it rather as my task to pave the way for a sympathetic understanding of modern algebra and to help its techniques, insofar as they are of general importance, to become generally acknowledged and to be integrated into the common store of knowledge of contemporary mathematicians. I am specifically talking to analysts whose approach and methods are directly opposite to modern algebraic techniques. My first way of characterizing the modern algebraic approach is to compare it with the traditional approach. It will not be my task to tell you of the many fine and meaningful results which have been achieved in algebra by modern methods over the last few years. I will only talk about questions of methodological and general interest. Among the wealth of the more recent investigations I will single out a few concepts to illustrate important methodological ideas. I hope you will forgive me if I concentrate on topics with which I am particularly familiar through my own interests and work. Nevertheless, I hope I will be able to complete my mission in a satisfactory way and will not be too one-sided in my presentation. 1. In the past, algebra, as well as all other mathematical disciplines involving computat ions, utilized exclusively the real and sometimes also the complex numbers. Today we use abstract fields and even integral domains and rings. What good is this? It would seem irrelevant whether, in deriving theorems on linear systems of equations, one thinks of the coefficients and variables as complex numbers or as elements of an abstract field. This being so, why should one think it important to operate in an abstract setting? There are two reasons for this. One's aim is the greatest possible generality of content. The more general the underlying initial assumptions the more comprehens ive the theory, the larger its realm of applications. In addition to the intrinsic satisfaction associated with maximal generality, there are other advantages. Take the theory of divisibility in an ordinary integral domain, that is, one with a euclidean algorithm. This theory is valid for the integers (factorization of integers into primes), for polynomials in one variable with coefficients in a field (decomposition of a polynomial into irreducible polynomials), and for the elementary divisors of matrices. Another example. Let k be an abstract ground field. If we show that the roots of the equation x" e = 0, where e is the unit element of k, form a cyclic group, then we will have proved, among other things, the existence of a primitive root modulo a prime p (by applying our result to the residue class field of the integers modulo the prime p). There are many such examples. I shall discuss one such e x a m p l e t h a t of an i d e a l i n some detail below. Thus, owing to its abstract foundation, modern

Character sums and primitive roots in algebraic number fields

Let ν denote a totally positive integer of an algebraic number fieldK such that ν is a least primitive root modulo a prime ideal $$\mathfrak{p}$$ ofK, least in the sense that its normNν is minimal. One of the simplest questions that presents itself is that of the order of magnitude ofNν in comparison toN $$\mathfrak{p}$$ . In the present paper the following bound is shown: $$N_\nu 0.$$ The proof of this result is based on deep estimates for certain character sums inK.

Character sums in algebraic number fields

The Pólya-Vinogradov inequality is generalized to arbitrary algebraic number fields K of finite degree over the rationals. The proof makes use of Siegel's summation formula and requires results about Hecke's zeta-functions with Grössencharacters. One application is to the problem of estimating a least totally positive primitive root modulo a prime ideal of K, least in the sense that its norm is minimal.

Lifting group homomorphisms

If a knot K has Alexander polynomial different from 1, then its knot group, G maps onto some metacyclic group, Zr§Zp. We show that in that case, it also has a homomorphism onto a split extension of a free abelian group of rank p — 1 by Zr§Zp, and hence also onto a split extension of a direct sum of p — 1 cyclic groups of order s by the metacyclic group. In many cases, (such as if s is coprime with p), this group can be specified exactly. Otherwise there are a finite number of possibilities. A special case is Perko's result that a homomorphism of a knot group onto S3 = Z2^Z?) lifts to S4 = Z 2ίS)Z3<$(Z2 Θ Z2). As an application we obtain information about the derived series of G. In a final section it is shown how to associate a rational polynomial invariant to every metacyclic representation. 1. Lifting metacyclic representations. Let p be a prime, r a divisor of p — 1 and β a primitive rth root modulo p. Let E — Zj§Zp — (Y, S: γr zn sp = 19Y'1SY= Sβ). Up to isomorphism, the group is independent of β. Let G be the knot group of a knot, AT, in the 3-sphere, S3, and let φ be a homomoφhism of G onto E which takes a meridian, m, of K to YaSh. Then g.c.d. (α, r) — 1, since G is generated by conjugates of m. Setting X — YaSb and eliminating 7, we obtain a presentation

On the Least Primitive Root Modulo p 2

Bulletin of the London Mathematical SocietyVolume 6, Issue 1 p. 42-46 Notes and papers On the Least Primitive Root Modulo p2 S. D. Cohen, S. D. Cohen Department of Mathematics, University Gardens, Glasgow, G12 8QWSearch for more papers by this authorR. W. K. Odoni, R. W. K. Odoni Department of Mathematics, University Gardens, Glasgow, G12 8QWSearch for more papers by this authorW. W. Stothers, W. W. Stothers Department of Mathematics, University Gardens, Glasgow, G12 8QWSearch for more papers by this author S. D. Cohen, S. D. Cohen Department of Mathematics, University Gardens, Glasgow, G12 8QWSearch for more papers by this authorR. W. K. Odoni, R. W. K. Odoni Department of Mathematics, University Gardens, Glasgow, G12 8QWSearch for more papers by this authorW. W. Stothers, W. W. Stothers Department of Mathematics, University Gardens, Glasgow, G12 8QWSearch for more papers by this author First published: March 1974 https://doi.org/10.1112/blms/6.1.42Citations: 6AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume6, Issue1March 1974Pages 42-46 RelatedInformation

The average of the least primitive root modulo $p^2$

The average of the least primitive root modulo $p^2$