1 Adeles over Q. 1.1 Absolute values

Size: px
Start display at page:

Download "1 Adeles over Q. 1.1 Absolute values"

Transcription

1 1 Adeles over Q 1.1 Absolute values Definition (Absolute value) An absolute value on a field F is a nonnegative real valued function on F which satisfies the conditions: (i) x = 0 if and only if x =0, (ii) xy = x y, (iii) x + y x + y, (triangle inequality) for all x, y F. If an absolute value on a field F satisfies the stronger condition x + y max ( x, y ), (1.1.2) then it is called a non-archimedean absolute value. If condition (1.1.2) fails for some x, y F, then is called an archimedean absolute value. It is always possible to define a trivial absolute value trivial on any field F where { 1, if x 0, x trivial = 0, otherwise. Since trivial is not very interesting, we shall usually exclude it in our discussions. Definition (Equivalence of absolute values) Two absolute values 1 and 2, defined on the same field F, are termed equivalent if there exists c > 0 such that x 1 = x c 2 for all x F. Example The field Q of rational numbers has the classical (and very ancient) archimedean absolute value which we denote by which is defined by { x, if x 0, x = x, if x < 0, (1.1.5) 1 in this web service

2 2 Adeles over Q for all x Q. For each prime p one may define the non-archimedean absolute value p as follows. Given x Q with x = p k m with p mn, and k Z, n we define x p = p k m = p k. (1.1.6) n The definition of p has the effect that the non-archimedean absolute values of numbers divisible by high powers of p become small. Theorem (Ostrowski) The only non-trivial absolute values on Q are those equivalent to the p or the ordinary absolute value. Proof See [Cassels, 1986], [Murty, 2002]. Theorem (Product formula) Let α Q with α 0. The absolute values v, given by (1.1.5), (1.1.6), satisfy the product formula α v =1 v where the product is taken over all v {, 2, 3, 5, 7, 11, 13,..., i.e., v = or v is a prime. Proof The proof is elementary and left to the reader. Definition (Finite and infinite primes) Following the modern tradition we shall call v =2, 3, 5, 7, 11, the finite primes and v = the infinite or archimedean prime. Henceforth, we shall adhere to the convention that v refers to an arbitrary prime v (with v finite or infinite), while p refers specifically to a finite prime. 1.2 The field Q p of p-adic numbers An absolute value on a field F allows us to define the notion of distance between two elements x, y F as x y. We may also introduce a topology on F where the basis of open sets consists of the open balls B r (a) with center a F and radius r > 0: B r (a) = { x x a < r. A sequence of elements x 1, x 2, x 3,... F is termed Cauchy provided x m x n 0, (m, n ). (1.2.1) A field F with a non-trivial absolute value is said to be complete if all Cauchy sequences of elements x 1, x 2, x 3,... F have the property that there in this web service

3 1.2 The field Q p of p-adic numbers 3 exists an element x F such that x n x 0asn, i.e., all Cauchy sequences converge. If a field F is not complete, it is possible to complete it by standard methods of analysis. In brief, one adjoins to the incomplete field F all the elements arising from equivalence classes of Cauchy sequences, where two Cauchy sequences {x 1, x 2,..., {y 1, y 2,... are equivalent if lim i x i y i =0. The original elements α F are then realized as the equivalence class of the constant Cauchy sequence {α, α, α,.... Addition, subtraction, and multiplication of the representatives {x i = {x 1, x 2,..., {y i = {y 1, y 2,... of two equivalence classes of Cauchy sequences are defined by {x i ±{y i = {x i ± y i, {x i {y i = {x i y i. The definition of division is the same, except one has to be careful to not divide by zero because in a Cauchy sequence {x 1, x 2, x 3,..., some of the x i may be 0. Happily, this is not a problem, because every Cauchy sequence is equivalent to a Cauchy sequence without any zero terms and we always choose such a representative for performing division. The sequence of quotients will be Cauchy, provided the Cauchy sequence by which we divide does not converge to zero. Definition (p-adic fields) Let p be a prime number. The completion of Q with respect to the p-adic absolute value p, defined by (1.1.6), is denoted as Q p and called the p-adic field. We now present two explicit constructions of Q p. Analytic construction of Q p : The first construction we present is based on the notion of Cauchy sequences. Let k < n be any two integers (positive or negative) and for each i satisfying k i n let 0 a i < p also be an integer. If we assume a k 0, then it easily follows from (1.1.6) that n a i p i = p k. (1.2.3) i=k p Fix k Z. An infinite sequence {a k, a k+1, a k+2,..., where a i {0, 1,..., p 1 for each i k, and a k 0, determines an infinite sequence x 1 = a k p k x 2 = a k p k + a k+1 p k+1 x 3 = a k p k + a k+1 p k+1 + a k+2 p k+2. in this web service

4 4 Adeles over Q of elements in Q. By (1.2.3) it is easy to see that the sequence x 1, x 2, x 3... is a Cauchy sequence. Formally, we may define lim x i = i a i p i, (with x i p = p k for all i =1, 2,...). i=k Let Z p denote the set of all elements x of the completed field Q p which satisfy x p 1. By (1.1.2) it easily follows that Z p must be a ring with maximal ideal { π = x Z p x p < 1. It is easy to check that π = p Z p. Every x Z p can be uniquely realized as the equivalence class of a Cauchy sequence of the form { a 0, a 0 + a 1 p, a 0 + a 1 p + a 2 p 2, a 0 + a 1 p + a 2 p 2 + a 3 p 3,... where 0 a i < p for i =0, 1, 2,... One may check this by first showing that every element of Z p contains a sequence consisting entirely of integers. Every integer may be expressed as a finite sum a a N p N. One then shows that for the sequence to be Cauchy, the digit a i must be eventually constant for each i. The ring Z p can thus be realized as the set of all sums of the type: a i p i (1.2.4) i=0 where 0 a i < p for each i 0. Suppose x Q p does not satisfy x p 1. Then we can multiply x by a suitable power p n with n > 0 so that p n x p 1. It immediately follows that the field Q p, can thus be realized as the set of all sums of the type: a i p i (1.2.5) i=k where 0 a i < p for each i k and k Z arbitrary. The actual mechanics of performing addition, subtraction, multiplication, and division in the field Q p is very similar to what we do in the field R where every element is of the form a k 10 k + a k 1 10 k 1 + (1.2.6) with 0 a i 9 for all i k. The main difference in Q p is that the expansion goes up instead of down as in (1.2.6). a k p k + a k+1 p k+1 + a k+2 p k+2 in this web service

5 1.2 The field Q p of p-adic numbers 5 Here is an example of multiplication in Q 5. Note that the multiplication and carrying procedures mimic the case of multiplication in R except that we move from left to right instead of right to left We give one more example of the type of infinite expansion that occurs in Q p which is analogous to the expansion 1 = that occurs in R. 3 Example Let a be an integer coprime to the prime p. Let f 1bea fixed integer. Then there exist integers ā, a 1, a 2,... such that 1 a = ā + a f p f + a f +1 p f +1 + a f +2 p f +2 + Q p where a a 1(modp f ) with 0 < ā < p f and 0 a i < p for i = f, f +1, f +2,... Since a 1 p = 1 it follows that a 1 must be in Z p and, thus, have an expansion of type (1.2.4). We require a (ā + a f p f + a f +1 p f +1 + ) =1 from which it easily follows that aā 1(modp f ). Note that p-adic expansions of p-adic numbers are always unique. This is not the case for decimal expansions of real numbers. For example: = Algebraic construction of Q p : Let A 1, A 2, A 3,... be an infinite set of groups, rings, or fields. We assume that for every pair of positive integers i, j with i > j there exists a homomorphism f i, j : A i A j. (1.2.8) in this web service

6 6 Adeles over Q Assume also that whenever i, j, k are positive integers satisfying i > j > k, that f i,k = f j,k f i, j. (1.2.9) Definition (Inverse limit) Let A 1, A 2, A 3,... be an infinite set of groups, rings, or fields. Assume that for all positive integers i > j that homomorphisms f i, j exist satisfying (1.2.8), (1.2.9). Then the inverse limit of the A i, denoted lim A i is defined to be the set of all infinite sequences (a 1, a 2, a 3,...) where a i A i for all i 1 and f i, j (a i )=a j for all i > j 1. The inverse limit inherits the algebraic structure of the sets A i. It will be either a group, ring or field. In the algebraic approach to the construction of Q p we first construct (using the inverse limit) the ring of p-adic integers, denoted Z p. The field Q p is then constructed as the field of quotients of Z p, consisting of all elements of the form a/b with a, b Z p and b 0. Note that Z p is an integral domain. Let p be a prime and let i be a positive integer. Then the set { A i := a 0 + a 1 p + a i 1 p i 1 0 al < p for all 0 l<i (1.2.11) determines a finite ring with p i elements which is canonically identified with the quotient ring (Z/p i Z). The algebraic operations are addition and multiplication modulo p i. For every i > j, we have the canonical homomorphism f i, j : A i A j defined by f i, j ( a0 + a 1 p + a i 1 p i 1) = a 0 + a 1 p + a j 1 p j 1, which simply drops off the tail end terms in the sum. It easily follows from Definition that an element of the inverse limit is a sequence of the form (a 0, a 0 + a 1 p, a 0 + a 1 p + a 2 p 2, a 0 + a 1 p + a 2 p 2 + a 3 p 3,...). Formally, we define the infinite sum i=0 a i p i to be the sequence above. Then ( / lim Z p i Z ) { = a i p i 0 a i < p for all i 0. (1.2.12) i=0 Definition (Ring of p-adic integers Z p ) Let p be a prime number. The ring of p-adic integers Z p is defined to be the inverse limit of finite rings given by (1.2.11). in this web service

7 1.3 Adeles and ideles over Q Adeles and ideles over Q The completion of Q with respect to the archimedean absolute value is just R which we also denote as Q. Formally, the ring of adeles over Q, denoted A Q, is a ring determined by the restricted product (relative to the subgroups Z p ) A Q = R p Q p, where restricted product (relative to the subgroups Z p ) means that all but finitely many of the components in the product are in Z p. Definition (Adeles) The ring of adeles over Q, denoted A Q, is defined by { A Q := {x, x 2, x 3,... x v Q v ( v ), x p Z p, ( but finitely many p). Given two adeles x = {x, x 2, x 3,..., x = {x, x 2, x 3,..., we define addition and multiplication (the ring operations) as follows x + x := {x + x, x 2 + x 2, x 3 + x 3,... x x := {x x, x 2 x 2, x 3 x 3,... Recall that a topological space X is called locally compact if every point of X has a compact neighborhood. For example, Q p is locally compact and Z p is compact. Furthermore, A Q can be made into a locally compact topological ring by taking as a basis for the topology all sets of the form U p S Z p where S is any finite set of primes containing, and U is any open subset in the product topology on the finite product v S Q v. (This follows the Tychonoff theorem, see [Munkres, 1975].) The ideles of Q are defined to be the multiplicative subgroup of A Q, denoted A Q. Definition (Ideles) The multiplicative group of ideles over Q, denoted A Q, is defined by A Q {{x :=, x 2,... A Q xv Q v ( v), x p Z p, ( but finitely many p). in this web service

8 8 Adeles over Q Here Z p denotes the multiplicative group of units of Z p. Clearly, u Z p if and only if u p =1. The ideles over Q also form a locally compact topological group with the basis of the topology consisting of the open sets U p S Z p where U is an open set in v S Q v and S is any finite set of primes containing. Here, the topology on the finite product v S Q v is the product topology. Warning: The topology of the ideles is not the topology induced from the adeles. It is quite different. Definition (Finite adeles) The ring of finite adeles over Q, denoted A finite, is defined by { A finite := {x 2, x 3,... x p Q p ( p < ), x p Z p, ( but finitely many p). There is a natural embedding of A finite into A Q given by {x 2, x 3,... {0, x 2, x 3,... Definition (Finite ideles) The group of finite ideles over Q, denoted A, is defined by finite A {{x := finite 2, x 3,... x p Q p ( p < ), x p Z p, ( but finitely many p). There is a natural embedding of A into A finite Q given by {x 2, x 3,... {1, x 2, x 3, Action of Q on the adeles and ideles The ring Q can be embedded in the adeles as follows. It is clear that for any fixed q Q that q v > 1 for only finitely many v. Thus q lies in Z p for all but finitely many p <. Let q Q. Then {q, q, q,... A Q. This is usually referred to as a diagonal embedding. It follows that Q may be considered as a subring of A Q.ViewingA Q and Q as additive groups, it is in this web service

9 1.4 Action of Q on the adeles and ideles 9 then natural to take the quotient Q\A Q. Another way to view this quotient is to define an additive action (denoted +) of Q on A Q by the formula q + x := {q + x, q + x 2, q + x 3,... for all x = {x, x 2, x 3,... A Q and all q Q. Here q + x v denotes addition in Q v. This is a continuous action and Q is a discrete subgroup of A Q in the sense that for each q Q, there is a subset U A Q, which is open in the topology on A Q, such that U Q = {q. We now introduce the notion of a fundamental domain for the action of an arbitrary group on an arbitrary set X. Definition (Fundamental domain) Let a group G act on a set X (on the left). A fundamental domain for this action is a subset D X which satisfies the following two properties: (1) For each x X, there exists d D and g G such that gx = d. (2) The choice of d in (1) is unique. Remarks A fundamental domain is precisely a choice of one point from each orbit of G. IfG\X is the quotient space with the quotient topology and π : X G\X is the quotient map, then the fundamental domain is the image of a section σ : G\X X. (This is a set theoretic section, it need not be continuous.) The construction of an explicit fundamental domain for the action of the additive group Q on the adele group A Q is equivalent to a generalization of the ancient Chinese remainder theorem. Theorem (Chinese Remainder Theorem) Let p 1, p 2,...p n be distinct primes. Let e 1, e 2,...,e n be positive integers and c 1, c 2,...,c n be arbitrary integers. Then the system of linear congruences x c 1 (mod p e 1 1 ) x c 2 (mod p e 2 2 ). x c n (mod p e n n ) has a unique solution x (mod p e 1 1 pe 2 2 pe n n ). Proof A simple proof can be obtained by explicitly constructing a solution to the system of linear congruences. Set N = p e 1 1 pe 2 2 pe n n. For each 1 i n define an integer u i by the condition N p e i i u i 1 (mod p e i i ). in this web service

10 10 Adeles over Q Then one easily checks that the element x c 1 N p e 1 1 u 1 + c 2 N p e 2 2 u c n N p e n n satisfies x c i (mod p e i i ) for all 1 i n. We leave the proof of uniqueness to the reader. Example Consider the system of linear congruences x 2(mod3 2 ) x 1(mod5 3 ) x 3(mod7). Then u 1 is defined by the congruence u 1 1(mod3 2 ), and u 1 =5. Similarly, u 2 1(mod5 3 ) and u 2 = 2, while u 3 1(mod7) and u 3 =3. It follows that x (mod ). A modern version of the Chinese Remainder Theorem (Theorem 1.4.2) can be given in terms of p-adic absolute values. Theorem (Weak approximation) Let p 1, p 2,...,p n be distinct primes. Let c i Q pi for each i =1, 2,...,n. Then for every ɛ>0, there exists an α Q such that α c i pi <ɛ for all 1 i n. Furthermore, α may be chosen so that the denominator, when written in lowest terms, is not divisible by any primes other than p 1,...,p n. Proof The general case follows easily from the case when c i Z pi for all i. As Z is dense in Z p, we may then replace c i by c i Z. At this point the statement reduces to the classical form, given in Theorem Proposition (Strong approximation for adeles) A fundamental domain DforQ\A Q is given by { D = {x, x 2, x 3,... 0 x < 1, x p Z p for all finite primes p =[0, 1) Z p. That is, we have p A Q = β Q {β + D, (disjoint union). u n in this web service

then it is called a non-archimedean absolute value. If condition (1.1.2) fails for some x, y F, then is called an archimedean absolute value.

then it is called a non-archimedean absolute value. If condition (1.1.2) fails for some x, y F, then is called an archimedean absolute value. CHAPTER I ADELES OVER Q 1.1 Absolute values Definition 1.1.1 (Absolute value) An absolute value on a field F is a nonnegative real valued function on F which satisfies the conditions: (i) x = 0 if and

More information

8 Complete fields and valuation rings

8 Complete fields and valuation rings 18.785 Number theory I Fall 2017 Lecture #8 10/02/2017 8 Complete fields and valuation rings In order to make further progress in our investigation of finite extensions L/K of the fraction field K of a

More information

Chapter 8. P-adic numbers. 8.1 Absolute values

Chapter 8. P-adic numbers. 8.1 Absolute values Chapter 8 P-adic numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.

More information

= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2

= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2 8. p-adic numbers 8.1. Motivation: Solving x 2 a (mod p n ). Take an odd prime p, and ( an) integer a coprime to p. Then, as we know, x 2 a (mod p) has a solution x Z iff = 1. In this case we can suppose

More information

a = a i 2 i a = All such series are automatically convergent with respect to the standard norm, but note that this representation is not unique: i<0

a = a i 2 i a = All such series are automatically convergent with respect to the standard norm, but note that this representation is not unique: i<0 p-adic Numbers K. Sutner v0.4 1 Modular Arithmetic rings integral domains integers gcd, extended Euclidean algorithm factorization modular numbers add Lemma 1.1 (Chinese Remainder Theorem) Let a b. Then

More information

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2018 57 5. p-adic Numbers 5.1. Motivating examples. We all know that 2 is irrational, so that 2 is not a square in the rational field Q, but that we can

More information

The p-adic numbers. Given a prime p, we define a valuation on the rationals by

The p-adic numbers. Given a prime p, we define a valuation on the rationals by The p-adic numbers There are quite a few reasons to be interested in the p-adic numbers Q p. They are useful for solving diophantine equations, using tools like Hensel s lemma and the Hasse principle,

More information

NOTES ON DIOPHANTINE APPROXIMATION

NOTES ON DIOPHANTINE APPROXIMATION NOTES ON DIOPHANTINE APPROXIMATION Jan-Hendrik Evertse January 29, 200 9 p-adic Numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics

More information

February 1, 2005 INTRODUCTION TO p-adic NUMBERS. 1. p-adic Expansions

February 1, 2005 INTRODUCTION TO p-adic NUMBERS. 1. p-adic Expansions February 1, 2005 INTRODUCTION TO p-adic NUMBERS JASON PRESZLER 1. p-adic Expansions The study of p-adic numbers originated in the work of Kummer, but Hensel was the first to truly begin developing the

More information

Part III. 10 Topological Space Basics. Topological Spaces

Part III. 10 Topological Space Basics. Topological Spaces Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.

More information

Valuations. 6.1 Definitions. Chapter 6

Valuations. 6.1 Definitions. Chapter 6 Chapter 6 Valuations In this chapter, we generalize the notion of absolute value. In particular, we will show how the p-adic absolute value defined in the previous chapter for Q can be extended to hold

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #5 09/19/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #5 09/19/2013 18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #5 09/19/2013 5.1 The field of p-adic numbers Definition 5.1. The field of p-adic numbers Q p is the fraction field of Z p. As a fraction field,

More information

Local Fields. Chapter Absolute Values and Discrete Valuations Definitions and Comments

Local Fields. Chapter Absolute Values and Discrete Valuations Definitions and Comments Chapter 9 Local Fields The definition of global field varies in the literature, but all definitions include our primary source of examples, number fields. The other fields that are of interest in algebraic

More information

NOTES IN COMMUTATIVE ALGEBRA: PART 2

NOTES IN COMMUTATIVE ALGEBRA: PART 2 NOTES IN COMMUTATIVE ALGEBRA: PART 2 KELLER VANDEBOGERT 1. Completion of a Ring/Module Here we shall consider two seemingly different constructions for the completion of a module and show that indeed they

More information

Solutions, Project on p-adic Numbers, Real Analysis I, Fall, 2010.

Solutions, Project on p-adic Numbers, Real Analysis I, Fall, 2010. Solutions, Project on p-adic Numbers, Real Analysis I, Fall, 2010. (24) The p-adic numbers. Let p {2, 3, 5, 7, 11,... } be a prime number. (a) For x Q, define { 0 for x = 0, x p = p n for x = p n (a/b),

More information

Algebraic Number Theory Notes: Local Fields

Algebraic Number Theory Notes: Local Fields Algebraic Number Theory Notes: Local Fields Sam Mundy These notes are meant to serve as quick introduction to local fields, in a way which does not pass through general global fields. Here all topological

More information

The p-adic Numbers. Akhil Mathew

The p-adic Numbers. Akhil Mathew The p-adic Numbers Akhil Mathew ABSTRACT These are notes for the presentation I am giving today, which itself is intended to conclude the independent study on algebraic number theory I took with Professor

More information

Profinite Groups. Hendrik Lenstra. 1. Introduction

Profinite Groups. Hendrik Lenstra. 1. Introduction Profinite Groups Hendrik Lenstra 1. Introduction We begin informally with a motivation, relating profinite groups to the p-adic numbers. Let p be a prime number, and let Z p denote the ring of p-adic integers,

More information

(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d

(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers

More information

The p-adic Numbers. Akhil Mathew. 4 May Math 155, Professor Alan Candiotti

The p-adic Numbers. Akhil Mathew. 4 May Math 155, Professor Alan Candiotti The p-adic Numbers Akhil Mathew Math 155, Professor Alan Candiotti 4 May 2009 Akhil Mathew (Math 155, Professor Alan Candiotti) The p-adic Numbers 4 May 2009 1 / 17 The standard absolute value on R: A

More information

Material covered: Class numbers of quadratic fields, Valuations, Completions of fields.

Material covered: Class numbers of quadratic fields, Valuations, Completions of fields. ALGEBRAIC NUMBER THEORY LECTURE 6 NOTES Material covered: Class numbers of quadratic fields, Valuations, Completions of fields. 1. Ideal class groups of quadratic fields These are the ideal class groups

More information

INVERSE LIMITS AND PROFINITE GROUPS

INVERSE LIMITS AND PROFINITE GROUPS INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological

More information

Number Theory, Algebra and Analysis. William Yslas Vélez Department of Mathematics University of Arizona

Number Theory, Algebra and Analysis. William Yslas Vélez Department of Mathematics University of Arizona Number Theory, Algebra and Analysis William Yslas Vélez Department of Mathematics University of Arizona O F denotes the ring of integers in the field F, it mimics Z in Q How do primes factor as you consider

More information

TEST CODE: PMB SYLLABUS

TEST CODE: PMB SYLLABUS TEST CODE: PMB SYLLABUS Convergence and divergence of sequence and series; Cauchy sequence and completeness; Bolzano-Weierstrass theorem; continuity, uniform continuity, differentiability; directional

More information

CLASS FIELD THEORY WEEK Motivation

CLASS FIELD THEORY WEEK Motivation CLASS FIELD THEORY WEEK 1 JAVIER FRESÁN 1. Motivation In a 1640 letter to Mersenne, Fermat proved the following: Theorem 1.1 (Fermat). A prime number p distinct from 2 is a sum of two squares if and only

More information

Chapter 1 The Real Numbers

Chapter 1 The Real Numbers Chapter 1 The Real Numbers In a beginning course in calculus, the emphasis is on introducing the techniques of the subject;i.e., differentiation and integration and their applications. An advanced calculus

More information

(1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define

(1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define Homework, Real Analysis I, Fall, 2010. (1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define ρ(f, g) = 1 0 f(x) g(x) dx. Show that

More information

NOTES ON FINITE FIELDS

NOTES ON FINITE FIELDS NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining

More information

Some Background Material

Some Background Material Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important

More information

Absolute Values and Completions

Absolute Values and Completions Absolute Values and Completions B.Sury This article is in the nature of a survey of the theory of complete fields. It is not exhaustive but serves the purpose of familiarising the readers with the basic

More information

Metric Spaces Math 413 Honors Project

Metric Spaces Math 413 Honors Project Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X X R such that for all x, y, z X: i) d(x, y) = d(y, x); ii) d(x, y) = 0 if and only

More information

Formal power series rings, inverse limits, and I-adic completions of rings

Formal power series rings, inverse limits, and I-adic completions of rings Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely

More information

Krull Dimension and Going-Down in Fixed Rings

Krull Dimension and Going-Down in Fixed Rings David Dobbs Jay Shapiro April 19, 2006 Basics R will always be a commutative ring and G a group of (ring) automorphisms of R. We let R G denote the fixed ring, that is, Thus R G is a subring of R R G =

More information

Standard forms for writing numbers

Standard forms for writing numbers Standard forms for writing numbers In order to relate the abstract mathematical descriptions of familiar number systems to the everyday descriptions of numbers by decimal expansions and similar means,

More information

ZEROES OF INTEGER LINEAR RECURRENCES. 1. Introduction. 4 ( )( 2 1) n

ZEROES OF INTEGER LINEAR RECURRENCES. 1. Introduction. 4 ( )( 2 1) n ZEROES OF INTEGER LINEAR RECURRENCES DANIEL LITT Consider the integer linear recurrence 1. Introduction x n = x n 1 + 2x n 2 + 3x n 3 with x 0 = x 1 = x 2 = 1. For which n is x n = 0? Answer: x n is never

More information

Math 210B. Artin Rees and completions

Math 210B. Artin Rees and completions Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an A-module. In class we defined the I-adic completion of M to be M = lim M/I n M. We will soon show

More information

Metric Spaces Math 413 Honors Project

Metric Spaces Math 413 Honors Project Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X X R such that for all x, y, z X: i) d(x, y) = d(y, x); ii) d(x, y) = 0 if and only

More information

Problem 1A. Use residues to compute. dx x

Problem 1A. Use residues to compute. dx x Problem 1A. A non-empty metric space X is said to be connected if it is not the union of two non-empty disjoint open subsets, and is said to be path-connected if for every two points a, b there is a continuous

More information

Lecture Notes. Advanced Discrete Structures COT S

Lecture Notes. Advanced Discrete Structures COT S Lecture Notes Advanced Discrete Structures COT 4115.001 S15 2015-01-13 Recap Divisibility Prime Number Theorem Euclid s Lemma Fundamental Theorem of Arithmetic Euclidean Algorithm Basic Notions - Section

More information

P-adic Functions - Part 1

P-adic Functions - Part 1 P-adic Functions - Part 1 Nicolae Ciocan 22.11.2011 1 Locally constant functions Motivation: Another big difference between p-adic analysis and real analysis is the existence of nontrivial locally constant

More information

p-adic Analysis Compared to Real Lecture 1

p-adic Analysis Compared to Real Lecture 1 p-adic Analysis Compared to Real Lecture 1 Felix Hensel, Waltraud Lederle, Simone Montemezzani October 12, 2011 1 Normed Fields & non-archimedean Norms Definition 1.1. A metric on a non-empty set X is

More information

Contents Ordered Fields... 2 Ordered sets and fields... 2 Construction of the Reals 1: Dedekind Cuts... 2 Metric Spaces... 3

Contents Ordered Fields... 2 Ordered sets and fields... 2 Construction of the Reals 1: Dedekind Cuts... 2 Metric Spaces... 3 Analysis Math Notes Study Guide Real Analysis Contents Ordered Fields 2 Ordered sets and fields 2 Construction of the Reals 1: Dedekind Cuts 2 Metric Spaces 3 Metric Spaces 3 Definitions 4 Separability

More information

Math 676. A compactness theorem for the idele group. and by the product formula it lies in the kernel (A K )1 of the continuous idelic norm

Math 676. A compactness theorem for the idele group. and by the product formula it lies in the kernel (A K )1 of the continuous idelic norm Math 676. A compactness theorem for the idele group 1. Introduction Let K be a global field, so K is naturally a discrete subgroup of the idele group A K and by the product formula it lies in the kernel

More information

Some topics in analysis related to Banach algebras, 2

Some topics in analysis related to Banach algebras, 2 Some topics in analysis related to Banach algebras, 2 Stephen Semmes Rice University... Abstract Contents I Preliminaries 3 1 A few basic inequalities 3 2 q-semimetrics 4 3 q-absolute value functions 7

More information

FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.

FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0

More information

THE P-ADIC NUMBERS AND FINITE FIELD EXTENSIONS OF Q p

THE P-ADIC NUMBERS AND FINITE FIELD EXTENSIONS OF Q p THE P-ADIC NUMBERS AND FINITE FIELD EXTENSIONS OF Q p EVAN TURNER Abstract. This paper will focus on the p-adic numbers and their properties. First, we will examine the p-adic norm and look at some of

More information

LECTURE NOTES, WEEK 7 MATH 222A, ALGEBRAIC NUMBER THEORY

LECTURE NOTES, WEEK 7 MATH 222A, ALGEBRAIC NUMBER THEORY LECTURE NOTES, WEEK 7 MATH 222A, ALGEBRAIC NUMBER THEORY MARTIN H. WEISSMAN Abstract. We discuss the connection between quadratic reciprocity, the Hilbert symbol, and quadratic class field theory. We also

More information

MATH 117 LECTURE NOTES

MATH 117 LECTURE NOTES MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set

More information

Algebraic function fields

Algebraic function fields Algebraic function fields 1 Places Definition An algebraic function field F/K of one variable over K is an extension field F K such that F is a finite algebraic extension of K(x) for some element x F which

More information

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of

More information

MEASURABLE DYNAMICS OF SIMPLE p-adic POLYNOMIALS JOHN BRYK AND CESAR E. SILVA

MEASURABLE DYNAMICS OF SIMPLE p-adic POLYNOMIALS JOHN BRYK AND CESAR E. SILVA MEASURABLE DYNAMICS OF SIMPLE p-adic POLYNOMIALS JOHN BRYK AND CESAR E. SILVA 1. INTRODUCTION. The p-adic numbers have many fascinating properties that are different from those of the real numbers. These

More information

Part III. x 2 + y 2 n mod m

Part III. x 2 + y 2 n mod m Part III Part III In this, the final part of the course, we will introduce the notions of local and global viewpoints of number theory, which began with the notion of p-adic numbers. (p as usual denote

More information

GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.

GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,

More information

p-adic fields Chapter 7

p-adic fields Chapter 7 Chapter 7 p-adic fields In this chapter, we study completions of number fields, and their ramification (in particular in the Galois case). We then look at extensions of the p-adic numbers Q p and classify

More information

MASTERS EXAMINATION IN MATHEMATICS SOLUTIONS

MASTERS EXAMINATION IN MATHEMATICS SOLUTIONS MASTERS EXAMINATION IN MATHEMATICS PURE MATHEMATICS OPTION SPRING 010 SOLUTIONS Algebra A1. Let F be a finite field. Prove that F [x] contains infinitely many prime ideals. Solution: The ring F [x] of

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #7 09/26/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #7 09/26/2013 18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #7 09/26/2013 In Lecture 6 we proved (most of) Ostrowski s theorem for number fields, and we saw the product formula for absolute values on

More information

7 Lecture 7: Rational domains, Tate rings and analytic points

7 Lecture 7: Rational domains, Tate rings and analytic points 7 Lecture 7: Rational domains, Tate rings and analytic points 7.1 Introduction The aim of this lecture is to topologize localizations of Huber rings, and prove some of their properties. We will discuss

More information

1 Take-home exam and final exam study guide

1 Take-home exam and final exam study guide Math 215 - Introduction to Advanced Mathematics Fall 2013 1 Take-home exam and final exam study guide 1.1 Problems The following are some problems, some of which will appear on the final exam. 1.1.1 Number

More information

Places of Number Fields and Function Fields MATH 681, Spring 2018

Places of Number Fields and Function Fields MATH 681, Spring 2018 Places of Number Fields and Function Fields MATH 681, Spring 2018 From now on we will denote the field Z/pZ for a prime p more compactly by F p. More generally, for q a power of a prime p, F q will denote

More information

The Completion of a Metric Space

The Completion of a Metric Space The Completion of a Metric Space Let (X, d) be a metric space. The goal of these notes is to construct a complete metric space which contains X as a subspace and which is the smallest space with respect

More information

RINGS: SUMMARY OF MATERIAL

RINGS: SUMMARY OF MATERIAL RINGS: SUMMARY OF MATERIAL BRIAN OSSERMAN This is a summary of terms used and main results proved in the subject of rings, from Chapters 11-13 of Artin. Definitions not included here may be considered

More information

Course MA2C02, Hilary Term 2013 Section 9: Introduction to Number Theory and Cryptography

Course MA2C02, Hilary Term 2013 Section 9: Introduction to Number Theory and Cryptography Course MA2C02, Hilary Term 2013 Section 9: Introduction to Number Theory and Cryptography David R. Wilkins Copyright c David R. Wilkins 2000 2013 Contents 9 Introduction to Number Theory 63 9.1 Subgroups

More information

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................

More information

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.

More information

Math 109 HW 9 Solutions

Math 109 HW 9 Solutions Math 109 HW 9 Solutions Problems IV 18. Solve the linear diophantine equation 6m + 10n + 15p = 1 Solution: Let y = 10n + 15p. Since (10, 15) is 5, we must have that y = 5x for some integer x, and (as we

More information

Chapter 5. Modular arithmetic. 5.1 The modular ring

Chapter 5. Modular arithmetic. 5.1 The modular ring Chapter 5 Modular arithmetic 5.1 The modular ring Definition 5.1. Suppose n N and x, y Z. Then we say that x, y are equivalent modulo n, and we write x y mod n if n x y. It is evident that equivalence

More information

OSTROWSKI S THEOREM FOR Q(i)

OSTROWSKI S THEOREM FOR Q(i) OSTROWSKI S THEOREM FOR Q(i) KEITH CONRAD We will extend Ostrowki s theorem from Q to the quadratic field Q(i). On Q, every nonarchimedean absolute value is equivalent to the p-adic absolute value for

More information

MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM

MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:00-10:00 PM Basic Questions 1. Compute the factor group Z 3 Z 9 / (1, 6). The subgroup generated by (1, 6) is

More information

MATH 361: NUMBER THEORY FOURTH LECTURE

MATH 361: NUMBER THEORY FOURTH LECTURE MATH 361: NUMBER THEORY FOURTH LECTURE 1. Introduction Everybody knows that three hours after 10:00, the time is 1:00. That is, everybody is familiar with modular arithmetic, the usual arithmetic of the

More information

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12 MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

More information

1 Absolute values and discrete valuations

1 Absolute values and discrete valuations 18.785 Number theory I Lecture #1 Fall 2015 09/10/2015 1 Absolute values and discrete valuations 1.1 Introduction At its core, number theory is the study of the ring Z and its fraction field Q. Many questions

More information

NUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z:

NUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z: NUMBER SYSTEMS Number theory is the study of the integers. We denote the set of integers by Z: Z = {..., 3, 2, 1, 0, 1, 2, 3,... }. The integers have two operations defined on them, addition and multiplication,

More information

THE REAL NUMBERS Chapter #4

THE REAL NUMBERS Chapter #4 FOUNDATIONS OF ANALYSIS FALL 2008 TRUE/FALSE QUESTIONS THE REAL NUMBERS Chapter #4 (1) Every element in a field has a multiplicative inverse. (2) In a field the additive inverse of 1 is 0. (3) In a field

More information

x = π m (a 0 + a 1 π + a 2 π ) where a i R, a 0 = 0, m Z.

x = π m (a 0 + a 1 π + a 2 π ) where a i R, a 0 = 0, m Z. ALGEBRAIC NUMBER THEORY LECTURE 7 NOTES Material covered: Local fields, Hensel s lemma. Remark. The non-archimedean topology: Recall that if K is a field with a valuation, then it also is a metric space

More information

The Mathematica Journal p-adic Arithmetic

The Mathematica Journal p-adic Arithmetic The Mathematica Journal p-adic Arithmetic Stany De Smedt The p-adic numbers were introduced by K. Hensel in 1908 in his book Theorie der algebraïschen Zahlen, Leipzig, 1908. In this article we present

More information

ECEN 5022 Cryptography

ECEN 5022 Cryptography Elementary Algebra and Number Theory University of Colorado Spring 2008 Divisibility, Primes Definition. N denotes the set {1, 2, 3,...} of natural numbers and Z denotes the set of integers {..., 2, 1,

More information

Part II. Number Theory. Year

Part II. Number Theory. Year Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section I 1G 70 Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler

More information

MATH 2112/CSCI 2112, Discrete Structures I Winter 2007 Toby Kenney Homework Sheet 5 Hints & Model Solutions

MATH 2112/CSCI 2112, Discrete Structures I Winter 2007 Toby Kenney Homework Sheet 5 Hints & Model Solutions MATH 11/CSCI 11, Discrete Structures I Winter 007 Toby Kenney Homework Sheet 5 Hints & Model Solutions Sheet 4 5 Define the repeat of a positive integer as the number obtained by writing it twice in a

More information

Introduction to Dynamical Systems

Introduction to Dynamical Systems Introduction to Dynamical Systems France-Kosovo Undergraduate Research School of Mathematics March 2017 This introduction to dynamical systems was a course given at the march 2017 edition of the France

More information

A CONSTRUCTION FOR ABSOLUTE VALUES IN POLYNOMIAL RINGS. than define a second approximation V 0

A CONSTRUCTION FOR ABSOLUTE VALUES IN POLYNOMIAL RINGS. than define a second approximation V 0 A CONSTRUCTION FOR ABSOLUTE VALUES IN POLYNOMIAL RINGS by SAUNDERS MacLANE 1. Introduction. An absolute value of a ring is a function b which has some of the formal properties of the ordinary absolute

More information

Rings. Chapter 1. Definition 1.2. A commutative ring R is a ring in which multiplication is commutative. That is, ab = ba for all a, b R.

Rings. Chapter 1. Definition 1.2. A commutative ring R is a ring in which multiplication is commutative. That is, ab = ba for all a, b R. Chapter 1 Rings We have spent the term studying groups. A group is a set with a binary operation that satisfies certain properties. But many algebraic structures such as R, Z, and Z n come with two binary

More information

In N we can do addition, but in order to do subtraction we need to extend N to the integers

In N we can do addition, but in order to do subtraction we need to extend N to the integers Chapter The Real Numbers.. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {, 2, 3, }. In N we can do addition, but in order to do subtraction we need to extend

More information

A brief introduction to p-adic numbers

A brief introduction to p-adic numbers arxiv:math/0301035v2 [math.ca] 7 Jan 2003 A brief introduction to p-adic numbers Stephen Semmes Abstract In this short survey we look at a few basic features of p-adic numbers, somewhat with the point

More information

CHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998

CHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998 CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic

More information

A number that can be written as, where p and q are integers and q Number.

A number that can be written as, where p and q are integers and q Number. RATIONAL NUMBERS 1.1 Definition of Rational Numbers: What are rational numbers? A number that can be written as, where p and q are integers and q Number. 0, is known as Rational Example:, 12, -18 etc.

More information

Problem Set 2: Solutions Math 201A: Fall 2016

Problem Set 2: Solutions Math 201A: Fall 2016 Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that

More information

MATH Fundamental Concepts of Algebra

MATH Fundamental Concepts of Algebra MATH 4001 Fundamental Concepts of Algebra Instructor: Darci L. Kracht Kent State University April, 015 0 Introduction We will begin our study of mathematics this semester with the familiar notion of even

More information

Course 2BA1: Trinity 2006 Section 9: Introduction to Number Theory and Cryptography

Course 2BA1: Trinity 2006 Section 9: Introduction to Number Theory and Cryptography Course 2BA1: Trinity 2006 Section 9: Introduction to Number Theory and Cryptography David R. Wilkins Copyright c David R. Wilkins 2006 Contents 9 Introduction to Number Theory and Cryptography 1 9.1 Subgroups

More information

arxiv: v1 [math.mg] 5 Nov 2007

arxiv: v1 [math.mg] 5 Nov 2007 arxiv:0711.0709v1 [math.mg] 5 Nov 2007 An introduction to the geometry of ultrametric spaces Stephen Semmes Rice University Abstract Some examples and basic properties of ultrametric spaces are briefly

More information

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory Part V 7 Introduction: What are measures and why measurable sets Lebesgue Integration Theory Definition 7. (Preliminary). A measure on a set is a function :2 [ ] such that. () = 2. If { } = is a finite

More information

3. The Sheaf of Regular Functions

3. The Sheaf of Regular Functions 24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice

More information

Simple Abelian Topological Groups. Luke Dominic Bush Hipwood. Mathematics Institute

Simple Abelian Topological Groups. Luke Dominic Bush Hipwood. Mathematics Institute M A E NS G I T A T MOLEM UNIVERSITAS WARWICENSIS Simple Abelian Topological Groups by Luke Dominic Bush Hipwood supervised by Dr Dmitriy Rumynin 4th Year Project Submitted to The University of Warwick

More information

The group (Z/nZ) February 17, In these notes we figure out the structure of the unit group (Z/nZ) where n > 1 is an integer.

The group (Z/nZ) February 17, In these notes we figure out the structure of the unit group (Z/nZ) where n > 1 is an integer. The group (Z/nZ) February 17, 2016 1 Introduction In these notes we figure out the structure of the unit group (Z/nZ) where n > 1 is an integer. If we factor n = p e 1 1 pe, where the p i s are distinct

More information

Filters in Analysis and Topology

Filters in Analysis and Topology Filters in Analysis and Topology David MacIver July 1, 2004 Abstract The study of filters is a very natural way to talk about convergence in an arbitrary topological space, and carries over nicely into

More information

Computing a Lower Bound for the Canonical Height on Elliptic Curves over Q

Computing a Lower Bound for the Canonical Height on Elliptic Curves over Q Computing a Lower Bound for the Canonical Height on Elliptic Curves over Q John Cremona 1 and Samir Siksek 2 1 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7

More information

A BRIEF INTRODUCTION TO LOCAL FIELDS

A BRIEF INTRODUCTION TO LOCAL FIELDS A BRIEF INTRODUCTION TO LOCAL FIELDS TOM WESTON The purpose of these notes is to give a survey of the basic Galois theory of local fields and number fields. We cover much of the same material as [2, Chapters

More information

Quadratic reciprocity (after Weil) 1. Standard set-up and Poisson summation

Quadratic reciprocity (after Weil) 1. Standard set-up and Poisson summation (September 17, 010) Quadratic reciprocity (after Weil) Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ I show that over global fields (characteristic not ) the quadratic norm residue

More information

We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero

We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.

More information

On convergent power series

On convergent power series Peter Roquette 17. Juli 1996 On convergent power series We consider the following situation: K a field equipped with a non-archimedean absolute value which is assumed to be complete K[[T ]] the ring of

More information

On a Homoclinic Group that is not Isomorphic to the Character Group *

On a Homoclinic Group that is not Isomorphic to the Character Group * QUALITATIVE THEORY OF DYNAMICAL SYSTEMS, 1 6 () ARTICLE NO. HA-00000 On a Homoclinic Group that is not Isomorphic to the Character Group * Alex Clark University of North Texas Department of Mathematics

More information