Introduction to Algebraic Number Theory
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
APPLICATIONS of GALOIS THEORY 1. Finite Fields Let F Be a Finite Field
CHAPTER IX APPLICATIONS OF GALOIS THEORY 1. Finite Fields Let F be a finite field. It is necessarily of nonzero characteristic p and its prime field is the field with p r elements Fp.SinceFis a vector space over Fp,itmusthaveq=p elements where r =[F :Fp]. More generally, if E ⊇ F are both finite, then E has qd elements where d =[E:F]. As we mentioned earlier, the multiplicative group F ∗ of F is cyclic (because it is a finite subgroup of the multiplicative group of a field), and clearly its order is q − 1. Hence each non-zero element of F is a root of the polynomial Xq−1 − 1. Since 0 is the only root of the polynomial X, it follows that the q elements of F are roots of the polynomial Xq − X = X(Xq−1 − 1). Hence, that polynomial is separable and F consists of the set of its roots. (You can also see that it must be separable by finding its derivative which is −1.) We q may now conclude that the finite field F is the splitting field over Fp of the separable polynomial X − X where q = |F |. In particular, it is unique up to isomorphism. We have proved the first part of the following result. Proposition. Let p be a prime. For each q = pr, there is a unique (up to isomorphism) finite field F with |F | = q. Proof. We have already proved the uniqueness. Suppose q = pr, and consider the polynomial Xq − X ∈ Fp[X]. As mentioned above Df(X)=−1sof(X) cannot have any repeated roots in any extension, i.e. -
On the Cohen-Macaulay Modules of Graded Subrings
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 2, Pages 735{756 S 0002-9947(04)03562-7 Article electronically published on April 27, 2004 ON THE COHEN-MACAULAY MODULES OF GRADED SUBRINGS DOUGLAS HANES Abstract. We give several characterizations for the linearity property for a maximal Cohen-Macaulay module over a local or graded ring, as well as proofs of existence in some new cases. In particular, we prove that the existence of such modules is preserved when taking Segre products, as well as when passing to Veronese subrings in low dimensions. The former result even yields new results on the existence of finitely generated maximal Cohen-Macaulay modules over non-Cohen-Macaulay rings. 1. Introduction and definitions The notion of a linear maximal Cohen-Macaulay module over a local ring (R; m) was introduced by Ulrich [17], who gave a simple characterization of the Gorenstein property for a Cohen-Macaulay local ring possessing a maximal Cohen-Macaulay module which is sufficiently close to being linear, as defined below. A maximal Cohen-Macaulay module (abbreviated MCM module)overR is one whose depth is equal to the Krull dimension of the ring R. All modules to be considered in this paper are assumed to be finitely generated. The minimal number of generators of an R-module M, which is equal to the (R=m)-vector space dimension of M=mM, will be denoted throughout by ν(M). In general, the length of a finitely generated Artinian module M is denoted by `(M)(orby`R(M), if it is necessary to specify the ring acting on M). -
A Note on Presentation of General Linear Groups Over a Finite Field
Southeast Asian Bulletin of Mathematics (2019) 43: 217–224 Southeast Asian Bulletin of Mathematics c SEAMS. 2019 A Note on Presentation of General Linear Groups over a Finite Field Swati Maheshwari and R. K. Sharma Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India Email: [email protected]; [email protected] Received 22 September 2016 Accepted 20 June 2018 Communicated by J.M.P. Balmaceda AMS Mathematics Subject Classification(2000): 20F05, 16U60, 20H25 Abstract. In this article we have given Lie regular generators of linear group GL(2, Fq), n where Fq is a finite field with q = p elements. Using these generators we have obtained presentations of the linear groups GL(2, F2n ) and GL(2, Fpn ) for each positive integer n. Keywords: Lie regular units; General linear group; Presentation of a group; Finite field. 1. Introduction Suppose F is a finite field and GL(n, F) is the general linear the group of n × n invertible matrices and SL(n, F) is special linear group of n × n matrices with determinant 1. We know that GL(n, F) can be written as a semidirect product, GL(n, F)= SL(n, F) oF∗, where F∗ denotes the multiplicative group of F. Let H and K be two groups having presentations H = hX | Ri and K = hY | Si, then a presentation of semidirect product of H and K is given by, −1 H oη K = hX, Y | R,S,xyx = η(y)(x) ∀x ∈ X,y ∈ Y i, where η : K → Aut(H) is a group homomorphism. Now we summarize some literature survey related to the presentation of groups. -
Dirichlet Series Associated to Cubic Fields with Given Quadratic Resolvent 3
DIRICHLET SERIES ASSOCIATED TO CUBIC FIELDS WITH GIVEN QUADRATIC RESOLVENT HENRI COHEN AND FRANK THORNE s Abstract. Let k be a quadratic field. We give an explicit formula for the Dirichlet series P Disc(K) − , K | | where the sum is over isomorphism classes of all cubic fields whose quadratic resolvent field is iso- morphic to k. Our work is a sequel to [11] (see also [15]), where such formulas are proved in a more general setting, in terms of sums over characters of certain groups related to ray class groups. In the present paper we carry the analysis further and prove explicit formulas for these Dirichlet series over Q, and in a companion paper we do the same for quartic fields having a given cubic resolvent. As an application, we compute tables of the number of S3-sextic fields E with Disc(E) < X, | | for X ranging up to 1023. An accompanying PARI/GP implementation is available from the second author’s website. 1. Introduction A classical problem in algebraic number theory is that of enumerating number fields by discrim- inant. Let Nd±(X) denote the number of isomorphism classes of number fields K with deg(K)= d and 0 < Disc(K) < X. The quantity Nd±(X) has seen a great deal of study; see (for example) [10, 7, 23]± for surveys of classical and more recent work. It is widely believed that Nd±(X)= Cd±X +o(X) for all d 2. For d = 2 this is classical, and the case d = 3 was proved in 1971 work of Davenport and Heilbronn≥ [13]. -
The Fundamental System of Units for Cubic Number Fields
University of Wisconsin Milwaukee UWM Digital Commons Theses and Dissertations May 2020 The Fundamental System of Units for Cubic Number Fields Janik Huth University of Wisconsin-Milwaukee Follow this and additional works at: https://dc.uwm.edu/etd Part of the Other Mathematics Commons Recommended Citation Huth, Janik, "The Fundamental System of Units for Cubic Number Fields" (2020). Theses and Dissertations. 2385. https://dc.uwm.edu/etd/2385 This Thesis is brought to you for free and open access by UWM Digital Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of UWM Digital Commons. For more information, please contact [email protected]. THE FUNDAMENTAL SYSTEM OF UNITS FOR CUBIC NUMBER FIELDS by Janik Huth A Thesis Submitted in Partial Fulllment of the Requirements for the Degree of Master of Science in Mathematics at The University of Wisconsin-Milwaukee May 2020 ABSTRACT THE FUNDAMENTAL SYSTEM OF UNITS FOR CUBIC NUMBER FIELDS by Janik Huth The University of Wisconsin-Milwaukee, 2020 Under the Supervision of Professor Allen D. Bell Let K be a number eld of degree n. An element α 2 K is called integral, if the minimal polynomial of α has integer coecients. The set of all integral elements of K is denoted by OK . We will prove several properties of this set, e.g. that OK is a ring and that it has an integral basis. By using a fundamental theorem from algebraic number theory, Dirichlet's Unit Theorem, we can study the unit group × , dened as the set of all invertible elements OK of OK . -
On Field Γ-Semiring and Complemented Γ-Semiring with Identity
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 8(2018), 189-202 DOI: 10.7251/BIMVI1801189RA Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN 0354-5792 (o), ISSN 1986-521X (p) ON FIELD Γ-SEMIRING AND COMPLEMENTED Γ-SEMIRING WITH IDENTITY Marapureddy Murali Krishna Rao Abstract. In this paper we study the properties of structures of the semi- group (M; +) and the Γ−semigroup M of field Γ−semiring M, totally ordered Γ−semiring M and totally ordered field Γ−semiring M satisfying the identity a + aαb = a for all a; b 2 M; α 2 Γand we also introduce the notion of com- plemented Γ−semiring and totally ordered complemented Γ−semiring. We prove that, if semigroup (M; +) is positively ordered of totally ordered field Γ−semiring satisfying the identity a + aαb = a for all a; b 2 M; α 2 Γ, then Γ-semigroup M is positively ordered and study their properties. 1. Introduction In 1995, Murali Krishna Rao [5, 6, 7] introduced the notion of a Γ-semiring as a generalization of Γ-ring, ring, ternary semiring and semiring. The set of all negative integers Z− is not a semiring with respect to usual addition and multiplication but Z− forms a Γ-semiring where Γ = Z: Historically semirings first appear implicitly in Dedekind and later in Macaulay, Neither and Lorenzen in connection with the study of a ring. However semirings first appear explicitly in Vandiver, also in connection with the axiomatization of Arithmetic of natural numbers. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
The Axiom of Choice, Zorn's Lemma, and the Well
THE AXIOM OF CHOICE, ZORN'S LEMMA, AND THE WELL ORDERING PRINCIPLE The Axiom of Choice is a foundational statement of set theory: Given any collection fSi : i 2 Ig of nonempty sets, there exists a choice function [ f : I −! Si; f(i) 2 Si for all i 2 I: i In the 1930's, Kurt G¨odelproved that the Axiom of Choice is consistent (in the Zermelo-Frankel first-order axiomatization) with the other axioms of set theory. In the 1960's, Paul Cohen proved that the Axiom of Choice is independent of the other axioms. Closely following Van der Waerden, this writeup explains how the Axiom of Choice implies two other statements, Zorn's Lemma and the Well Ordering Prin- ciple. In fact, all three statements are equivalent, as is a fourth statement called the Hausdorff Maximality Principle. 1. Partial Order Let S be any set, and let P(S) be its power set, the set whose elements are the subsets of S, P(S) = fS : S is a subset of Sg: The proper subset relation \(" in P(S) satisfies two conditions: • (Partial Trichotomy) For any S; T 2 P(S), at most one of the conditions S ( T;S = T;T ( S holds, and possibly none of them holds. • (Transitivity) For any S; T; U 2 P(S), if S ( T and T ( U then also S ( U. The two conditions are the motivating example of a partial order. Definition 1.1. Let T be a set. Let ≺ be a relation that may stand between some pairs of elements of T . -
An Introduction to Skolem's P-Adic Method for Solving Diophantine
An introduction to Skolem's p-adic method for solving Diophantine equations Josha Box July 3, 2014 Bachelor thesis Supervisor: dr. Sander Dahmen Korteweg-de Vries Instituut voor Wiskunde Faculteit der Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam Abstract In this thesis, an introduction to Skolem's p-adic method for solving Diophantine equa- tions is given. The main theorems that are proven give explicit algorithms for computing bounds for the amount of integer solutions of special Diophantine equations of the kind f(x; y) = 1, where f 2 Z[x; y] is an irreducible form of degree 3 or 4 such that the ring of integers of the associated number field has one fundamental unit. In the first chapter, an introduction to algebraic number theory is presented, which includes Minkovski's theorem and Dirichlet's unit theorem. An introduction to p-adic numbers is given in the second chapter, ending with the proof of the p-adic Weierstrass preparation theorem. The theory of the first two chapters is then used to apply Skolem's method in Chapter 3. Title: An introduction to Skolem's p-adic method for solving Diophantine equations Author: Josha Box, [email protected], 10206140 Supervisor: dr. Sander Dahmen Second grader: Prof. dr. Jan de Boer Date: July 3, 2014 Korteweg-de Vries Instituut voor Wiskunde Universiteit van Amsterdam Science Park 904, 1098 XH Amsterdam http://www.science.uva.nl/math 3 Contents Introduction 6 Motivations . .7 1 Number theory 8 1.1 Prerequisite knowledge . .8 1.2 Algebraic numbers . 10 1.3 Unique factorization . 15 1.4 A geometrical approach to number theory . -
Section V.1. Field Extensions
V.1. Field Extensions 1 Section V.1. Field Extensions Note. In this section, we define extension fields, algebraic extensions, and tran- scendental extensions. We treat an extension field F as a vector space over the subfield K. This requires a brief review of the material in Sections IV.1 and IV.2 (though if time is limited, we may try to skip these sections). We start with the definition of vector space from Section IV.1. Definition IV.1.1. Let R be a ring. A (left) R-module is an additive abelian group A together with a function mapping R A A (the image of (r, a) being × → denoted ra) such that for all r, a R and a,b A: ∈ ∈ (i) r(a + b)= ra + rb; (ii) (r + s)a = ra + sa; (iii) r(sa)=(rs)a. If R has an identity 1R and (iv) 1Ra = a for all a A, ∈ then A is a unitary R-module. If R is a division ring, then a unitary R-module is called a (left) vector space. Note. A right R-module and vector space is similarly defined using a function mapping A R A. × → Definition V.1.1. A field F is an extension field of field K provided that K is a subfield of F . V.1. Field Extensions 2 Note. With R = K (the ring [or field] of “scalars”) and A = F (the additive abelian group of “vectors”), we see that F is a vector space over K. Definition. Let field F be an extension field of field K. -
Introduction to the P-Adic Space
Introduction to the p-adic Space Joel P. Abraham October 25, 2017 1 Introduction From a young age, students learn fundamental operations with the real numbers: addition, subtraction, multiplication, division, etc. We intuitively understand that distance under the reals as the absolute value function, even though we never have had a fundamental introduction to set theory or number theory. As such, I found this topic to be extremely captivating, since it almost entirely reverses the intuitive notion of distance in the reals. Such a small modification to the definition of distance gives rise to a completely different set of numbers under which the typical axioms are false. Furthermore, this is particularly interesting since the p-adic system commonly appears in natural phenomena. Due to the versatility of this metric space, and its interesting uniqueness, I find the p-adic space a truly fascinating development in algebraic number theory, and thus chose to explore it further. 2 The p-adic Metric Space 2.1 p-adic Norm Definition 2.1. A metric space is a set with a distance function or metric, d(p, q) defined over the elements of the set and mapping them to a real number, satisfying: arXiv:1710.08835v1 [math.HO] 22 Oct 2017 (a) d(p, q) > 0 if p = q 6 (b) d(p,p)=0 (c) d(p, q)= d(q,p) (d) d(p, q) d(p, r)+ d(r, q) ≤ The real numbers clearly form a metric space with the absolute value function as the norm such that for p, q R ∈ d(p, q)= p q . -
Local-Global Methods in Algebraic Number Theory
LOCAL-GLOBAL METHODS IN ALGEBRAIC NUMBER THEORY ZACHARY KIRSCHE Abstract. This paper seeks to develop the key ideas behind some local-global methods in algebraic number theory. To this end, we first develop the theory of local fields associated to an algebraic number field. We then describe the Hilbert reciprocity law and show how it can be used to develop a proof of the classical Hasse-Minkowski theorem about quadratic forms over algebraic number fields. We also discuss the ramification theory of places and develop the theory of quaternion algebras to show how local-global methods can also be applied in this case. Contents 1. Local fields 1 1.1. Absolute values and completions 2 1.2. Classifying absolute values 3 1.3. Global fields 4 2. The p-adic numbers 5 2.1. The Chevalley-Warning theorem 5 2.2. The p-adic integers 6 2.3. Hensel's lemma 7 3. The Hasse-Minkowski theorem 8 3.1. The Hilbert symbol 8 3.2. The Hasse-Minkowski theorem 9 3.3. Applications and further results 9 4. Other local-global principles 10 4.1. The ramification theory of places 10 4.2. Quaternion algebras 12 Acknowledgments 13 References 13 1. Local fields In this section, we will develop the theory of local fields. We will first introduce local fields in the special case of algebraic number fields. This special case will be the main focus of the remainder of the paper, though at the end of this section we will include some remarks about more general global fields and connections to algebraic geometry.