Euclidean Models of the p-adic Integers

Size: px
Start display at page:

Download "Euclidean Models of the p-adic Integers"

Transcription

1 Euclidean Models of the p-adic Integers Scott Zinzer December 12, Introduction Much of our visual perception is based in what seems to be standard Euclidean space; we can easily imagine perfectly straight lines and planes, and we can even create realistic drawings of the world we perceive using straight lines. Whether the physical universe is actually Euclidean in its geometry is open for debate, but it is hard to deny our reliance on Euclidean geometry for perceiving the space in which we immediately exist. Certainly, this is enforced by the prevalence of Euclidean geometry in the standard elementary mathematics education curriculum. So strong is our reliance on Euclidean geometry that even in the setting of general (potentially non-euclidean) spaces, we often attempt to create Euclidean pictures to help us better understand the spaces under consideration. Even the words we use in these general settings seem inspired by our experience with Euclidean spaces (e.g. balls, disks, and spheres in metric spaces). Kurt Hensel first discovered 1 the p-adic numbers in the late 19th century (here p is a prime number). The p-adic world provides a canvas on which algebra, analysis, and number theory interplay to create a landscape quite unlike anything expected from our immediate experience with Euclidean space. Topologically, the ring Z p of p-adic integers provides a mess of counterexamples to standard misconceptions in metric space topology (phrases such as open ball and closed ball become meaningless), although the average non number-theorist may never have a formal introduction to the p-adic integers or to the p-adic topology. There are several ways of motivating and constructing the p-adic numbers algebraically or analytically. In this paper, we will simply construct the p-adic integers from the rational integers (Z) by introducing the p-adic 1 or invented, depending on one s philosophical view of mathematics. 1

2 metric on Z and completing. Although most of the algebraic construction will be supressed, there is one striking fact worth mentioning. The ring of p-adic integers is a subring of the field of p-adic numbers, the field obtained by completing the rational numbers with respect to the p-adic absolute value and its induced metric. Ostrowski s theorem states that any (non-trivial) absolute value on Q is equivalent to the usual (real) absolute value or to some p-adic absolute value. In this way, the only completions of Q with respect to metrics inherited from non-trivial absolute values on Q are the field of real numbers (R) or some p-adic field. In this way, the p-adic numbers actually arise just as naturally from Q as does R. For many number-theoretical applications, the p-adic metrics are much more useful than the usual (real) absolute value, and we actually come to adopt several conventions in order to treat R on the same footing as the p-adic completions of Q. The primary goal of this paper is to attempt to describe methods of visualizing the p-adic integers within familiar Euclidean space. To do so, we seek subsets of n-dimensional Euclidean space that are homeomorphic to the p-adic integers that we can describe by an inductive construction. These subsets turn out to be fractal subsets of Euclidean space, and this perhaps helps to highlight just how novel the p-adic integers are in a topological sense. 1.1 Notation Z, Q, R, and C will denote the ring of rational integers, the field of rational numbers, the field of real numbers, and the field of complex numbers, as usual. p will always denote a prime number in Z + and will often be fixed in what follows. 2 Background 2.1 Ultrametric Spaces The standard introductory references for the following material include [2, 4]. The material in this first subsection is motivated especially by [6]. See also [5, 3]. Recall that metric on a set X is a function d : X X R satisfying 1. d(x, y) 0 for all x, y X. 2. d(x, y) = 0 if and only if x = y. 2

3 3. d(x, y) = d(y, x) for all x, y X. 4. d(x, y) d(x, z) + d(z, y) for all x, y, z X. If X is a set with metric d, then the pair (X, d) is called a metric space. Property 4 above of a metric d is often endearingly referred to as the triangle inequality, in analogy with the corresponding property of the lengths of the sides of Euclidean triangles. We strengthen property 4 to the following: 4. d(x, y) max{d(x, z), d(z, y)} for all x, y, z X. The new property 4 is called the strong triangle inequality (sometimes also the ultrametric inequality) 2. Clearly, any function d satisfying 4 also satisfies 4, and this leads us to our first definition: Definition 1. Let (X, d) be a metric space. The metric d is called an ultrametric 3 and (X, d) is called an ultrametric space if d satisfies the strong triangle inequality. Obviously, all of the properties of metric spaces carry over for ultrametric spaces, but the strong triangle inequality introduces several notable topological properties not holding in general for metric spaces. As usual, given an ultrametric space X, a point x X, and r > 0, the open ball of radius r centered at a is B r (x) = {y X : d(x, y) < r} and the closed ball of radius r centered at x is B r (x) = {y X : d(x, y) r} Our notation is not meant to imply that B r (a) is the topological closure of B r (a), and the terminology is not meant to imply openness or closedness with respect to the topology on X. When we refer simply to a ball in X, we mean a set of one of the above forms. Lemma 1. Let (X, d) be an ultrametric space. 2 For this reason, I will often refer to the usual triangle inequality as the weak triangle inequality. 3 In the setting of p-adic analysis, the ultrametric is sometimes also called non- Archimedean, although this term is most often reserved for the absolute value that arises from the p-adic valuation. Of course, this absolute value induces the p-adic metric. 3

4 1. Let x, y, z X with d(x, z) d(y, z). Then d(x, y) = max{d(x, z), d(y, z)} (all triangles are isosceles). 2. Any ball in X is both open and closed. 3. Any point of a ball may be its center. 4. If B 1 and B 2 are balls in X which are not disjoint, then B 1 B 2 or B 2 B 1. Proof. 1. Without loss of generality, suppose d(x, z) < d(y, z). Then But also, d(x, y) max{d(x, z), d(y, z)} = d(y, z) d(x, z) < d(y, z) max{d(x, y), d(x, z)} Thus, we must have d(y, z) d(x, y). 2. Let x X and r > 0. Of course, B r (x) is open and B r (x) is closed. Suppose y X \ B r (x). Then d(x, y) r. If d(y, z) < r, then d(y, z) < d(x, y). Thus d(x, z) = max{d(x, y), d(y, z)} = d(x, y) r This gives B r (y) X\B r (x), so B r (x) is closed. Finally, let y B r (x). If z B r (y), then d(z, x) max{d(z, y), d(x, y)} r so B r (y) B r (x) and B r (x) is open. 3. First consider a ball of the form B r (x) and let y B r (x). If z B r (y), then d(z, x) max{d(z, y), d(y, x)} < r so that B r (y) B r (x). A symmetrical argument gives the reverse containment, so B r (x) = B r (y). An identical argument works for a ball of the form B r (x). 4. If neither B 1 B 2 nor B 2 B 1, choose x B 1 \ B 2, y B 2 \ B 1 and z B 1 B 2. From the previous part, z is a center of both B 1 and B 2. Therefore d(z, x) > d(z, y) since x B 1 and y B 1 and d(z, x) < d(z, y) since y B 2 and x B 2. This is impossible. 4

5 We will need to discuss completions of ultrametric spaces in the next section, so we record the following: Lemma 2. Let (X, d) be an ultrametic space and (x n ) a sequence in X. Then (x n ) is Cauchy if and only if for all ε > 0 there is N > 0 so that d(x n, x n+1 ) < ε for all n N. Proof. Obviously any Cauchy sequence satisfies this property. Now let (x n ) be any sequence in X satisfying this property and let ε > 0. Choose N > 0 so that d(x n, x n+1 ) < ε for all n N. Now let m > n N. Then d(x m, x n ) max{d(x n, x n+1 ), d(x n+1, x n+2 ),, d(x m 1, x m )} < ε. 2.2 The p-adic Integers Let p Z be a fixed prime. Any element x Z \ {0} can be written in the form x = p e y, where y is relatively prime to p. By unique factorization in the integers, the exponent of p which appears in this factorization is unique. Definition 2. For x Z \ {0}, the p-adic valuation of x is the non-negative integer v p (x) such that x = p vp(x) y with y relatively prime to p. It is relatively straightforward to see that v p satisfies the following properties: 1. v p (x) 0 for all x Z \ {0}. 2. v p (xy) = v p (x) + v p (y) for all x, y Z \ {0}. 3. v p (x + y) min{v p (x), v p (y)} for all x, y Z \ {0}. We extend v p to all of Z by putting v p (0) = ; then v p : Z Z 0 { } still satisfies 1-3 above on all of Z. We use the p-adic valuation to define a metric on Z. Definition 3. The p-adic topology on Z is the metric topology with the p- adic metric d : Z Z R given by d(x, y) = p vp(x y) (where p = 0) 4. With the p-adic topology, Z is an ultrametric space. 4 In essence, the p-adic metric is measuring distance in terms of congruence modulo the prime p. Integers that are congruent modulo a high power of p have a difference with a large p-adic valuation, and hence are assigned a small distance by the p-adic metric. 5

6 Hereafter, we always endow Z with the p-adic topology. Notice that {d(x, y) : x, y Z} = {0} {p n : n 0} Definition 4. For a metric space (X, d), the metric d is called discrete if x 1, x 2,... X, y 1, y 2,... X with d(x 1, y 1 ) > d(x 2, y 2 ) > implies d(x n, y n ) 0. The p-adic metric on Z is discrete. All of the general properties of ultrametric spaces above hold for Z with the p-adic topology, but unfortunately, Z ( is not complete with respect to this topology. For instance, the sequence n k=0 pk) is p-adic Cauchy but does not converge p-adically in Z. n 0 Definition 5. The p-adic integers Z p is the completion of Z with respect to the p-adic topology. Then Z p is a complete ultrametric space and Z is dense in Z p, making Z p a separable ultrametric space. Note that we still have {d(x, y) : x, y Z p } = {0} {p n : n 0} so that the p-adic metric on Z p is discrete. In fact, N is dense in Z p and there is a particularly useful way of representing any element of Z p as a limit of a sequence in N. Recall that each element a N has a unique finite base-p expansion a = a 0 + a 1 p + a 2 p a N p N with a i {0,..., p 1}. Lemma 3. Let α Z p. Then α can be written uniquely as α = i 0 a i p i with a i {0,..., p 1} 5. We call this the p-adic expansion of α. From this result, it follows that Z p is uncountable. We use the p-adic expansion to prove the following. Lemma 4. Z p is sequentially compact, hence compact. 5 There are other possible choices for the p-adic digits a i, but {0,..., p 1} will suffice for our purposes. 6

7 Proof. Let (α n ) be a sequence in Z p. For each n, we write α n = i 0 a (n) i p i By the pigeonhole principle, there is b 0 {0,..., p 1} for which a (n) 0 = b 0 for infinitely many n. The collection of all such terms of (α n ) yields a subsequence (α 0n ), the terms of which all have b 0 as the first digit of their p-adic expansions. We now repeat this construction inductively to obtain a sequence of subsequences of (α n ), ((α kn ) n ) k with (α kn ) n a subsequence of (α k+1,n ) n and a p-adic integer b = k 0 b k p k such that for each k, every term of (α kn ) n agrees with b in its first k + 1 digits. Then the diagonal sequence (α kk ) is a subsequence of (α n ) which converges to b. It is easy to determine the distance between two p-adic integers if their p-adic digits are known. If the first n digits of x, y Z p are identical, then the first n digits of x y are all 0, which means that p n divides x y, so that d(x, y) p n. Consider now any closed ball in Z p, say B ε (x). Then there is n 0 so that B ε (x) = B 1/p n(x). Now let y Z have the same initial n p-adic digits as x (y is given by truncating the p-adic expansion of x). Then B 1/p n(y) = B ε (x) and we can apply a similar argument to an open ball in Z p. Therefore, Lemma 5. The collection of balls in Z p is countable. Thus, Z p is a second countable metric space. In light of the above, we will always write a ball in Z p as a ball with a radius of the form p n for some n centered at a Z with 0 a p n 1. We now change notation 6 and let a + p n Z p = B 1/p n(a) The properties of balls in ultrametric spaces implies p 1 a + p n Z p = (a + bp n ) + p n+1 Z p b=0 6 This change reflects the algebraic structure of Z p. 7

8 and in particular, Finally, we record p 1 Z p = b + pz p b=0 Lemma 6. Z p is totally disconnected. Proof. Let a Z p and let A Z p be any connected subset containing a. Suppose there is b A with b a. Choose 0 < ε < d(a, b). Then B ε (a) is both open and closed in Z p, hence B ε (a) A, A is both open and closed in A. This is impossible, so it must be that A = {a}. This result implies that Z p is not locally Euclidean, so is certainly not a real manifold (although it is Hausdorff and second countable) 7. For this paper, we are primarily concerned with Z p as a topological space (a metric space), so we neglect many of the algebraic properties of Z p, although some of these properties have already come into play behind the scenes. For the sake of completeness we cite some useful algebraic facts. Fact 1. The following are true: 1. Z p is a (topological) ring and the addition and multiplication of p-adic integers are the only continuous operations on Z p extending addition and multiplication on Z. The ring operations can be defined on p-adic expansions and are described by polynomials in finitely many variables. 2. Z p is a principal ideal domain and its nontrivial ideals are precisely the ideals generated by p k. 3. Z p is a local ring with unique maximal ideal pz p. Therefore, Z p = Z p \ pz p. 4. Z p /pz p = Z/pZ. Taking for granted that Z p is a ring, we observe that for any x, y Z p d(px, py) = p vp(px py) = p vp(p(x y)) = p vp(p) vp(x y) = p 1 p vp(x y) = p 1 d(x, y) 7 Although the title of this course is The Geometry and Topology of Manifolds, I still hoped this topic would be appreciated and would be of general interest. 8

9 so that multiplication by p is a contraction mapping. important in the following section. This fact will be 3 Euclidean Models of Z p The material in this section is inspired by the related material in [5] and [3]. The paper [1] is also a source of motivation. The ultimate goal is to embed Z p topologically into Euclidean space. The fact that this can be done is perhaps a bit surprising. We remark however, that although we can topologically embed Z p into Euclidean space, we cannot do so while preserving algebraic structure. Definition 6. A subset E R n which is homeomorphic to Z p is called a Euclidean model of Z p. A subset E R which is homeomorphic to Z p is called a linear model of Z p. In what follows, we will describe some Euclidean models of Z p and iterative constructions to visualize them. To find such subsets of R n, the general approach will be to exploit the decomposition of Z p as a disjoint union of open balls of the form a + pz p with a {0,..., p 1} and to use the fact that multiplication by p is a contraction mapping (so that pz p looks like a shrunken copy of Z p ). 3.1 Cantor Sets We begin with a somewhat unsatisfactory Euclidean model of Z p ; namely, we show that there is essentially only one family of linear models of Z p. However, there is one interesting fact that results from this consideration. Recall that the (classical) Cantor set is a subset of the unit inverval I = [0, 1] R obtained from the following iterative construction. From C 0 = I = [0, 1], delete the open middle third (1/3, 2/3) to obtain C 1 = [0, 1/3] [2/3, 1], a union of two closed intervals. To obtain C n+1 from C n, where C n is a union of closed intervals, delete the open middle third of each closed interval. Then each C n consists of 2 n closed intervals of length 3 n (so is compact) and C n+1 C n. The Cantor set C is C = C n [0, 1] with the subset (metric) topology inherited from R. Lemma 7. The Cantor set C has the following properties. 9

10 1. C is compact. 2. C is uncountable. 3. C is totally disconnected. 4. C is perfect, meaning it is closed and does not contain isolated points. Properties 1-3 should sound familiar; they are topological properties of Z p that we highlighted in the previous section. In fact, there is a (classification) theorem about subsets of R satisfying the above properties. Theorem 1. Any compact, perfect, totally disconnected subset of R is homeomorphic to the Cantor set. This theorem implies that any linear model of Z p must be homeomorphic to the Cantor set. We now show that the classical Cantor set C is a linear model of Z 2, the 2-adic integers, and then describe Cantor-like subsets which are linear models of Z p. To describe points of C, we adopt a base-3 numeration system with digits {0, 1, 2} for points in [0, 1]. Then the set C n in the iterative construction of C is the subset of [0, 1] whose first n digits in this base-3 expansion are not 1. Therefore, C consists of all numbers in [0, 1] with base-3 expansions c i 3 i i 1 with c i {0, 1}, and any element of C can be uniquely expressed in this form. This looks promising, since there are only two choices for 2-adic digits for 2-adic expansions of elements of Z 2. We define Φ : Z 2 C via Φ : i 0 a i 2 i (2a i )3 (i+1) i 0 This certainly gives a bijection Z 2 C by the uniqueness of the expansions, but more is true. Lemma 8. Φ : Z 2 C is a homeomorphism. Proof. Let x, y Z 2 with p-adic expansions x = i 0 x i 2 i, y = i 0 y i 2 i Then d(x, y) 2 n if and only if x i = y i for all i n. But this occurs if and only if 2x i = 2y i for all i n, which occurs if and only of Φ(x) and Φ(y) are in the same component of C n, which is equivalent to Φ(x) Φ(y) 3 n. This gives the continuity of both Φ and Φ 1. 10

11 For linear models of Z p, we generalize the construction of the Cantor set slightly. For a prime number p, the generalized p-cantor set C (p) is a subset of the unit inverval I = [0, 1] R obtained from the following iterative construction. Divide C (p) 0 = I = [0, 1] into 2p 1 equal subintervals and delete every second open interval to obtain C (p) 1 To obtain C (p), where C (p), a union of closed intervals. n+1 from C(p) n n is a union of closed intervals, divide each closed interval in 2p 1 equal subintervals and delete every second open interval. Then C (p) = C n (p) is uncountable, compact, and perfect. If we adopt a base-(2p 1) numeration system with digits {0, 1,..., 2p 2} for [0, 1], then C n is the subset of [0, 1] whose first n digits are even. Then C (p) consists of all numbers in [0, 1] with expansions c i (2p 1) i i 1 with c i {0, 2, 4,... 2p 2}, and any element of C (p) can be uniquely expressed in this form. Lemma 9. For any p, the map Φ (p) : Z p C (p) given by Φ (p) : i 0 a i p i (2a i )(2p 1) (i+1) i 0 is a homeomorphism. The proof of this result is analogous to the proof for Z 2 and C: two p-adic integers are close if and only if their images under Φ (p) lie in the same connected component of some approximating set C n (p), which occurs if and only if the images are close with respect to the Euclidean metric on R. Since C and C (p) are homeomorphic, we obtain the following mildly surprising result. Lemma 10. Z p and Z q are homeomorphic for all p, q. In particular, the subsets of R n we obtain in the following section are all homeomorphic. 3.2 Non-linear Euclidean Models of Z p We now describe subsets of R n (n 2) which are homeomorphic to Z p. For fixed p, let ν : {0,..., p 1} R n be an injective map, and put 11

12 S = ν({0,..., p 1}) R n. Define ψ b : Z p R n by ψ b : i 0 a i p i i 0 (b 1) b i+1 ν(a i) Then ψ b (Z p ) is contained in the convex hull of S. Recall that Z p = p 1 a=0 a + pz p Then ψ b (Z p ) = p 1 a=0 = v S ( (b 1) b ( (b 1) b ν(a) + 1 ) b ψ b(z p ) v + 1 ) b ψ b(z p ) i.e. ψ b (Z p ) is a union of translates of dilated copies of ψ b (Z p ). When b is chosen large enough, in fact ψ b is injective and ψ b (Z p ) is also a disjoint union of self-similar images, i.e. a fractal subset of R n. Let F = ψ b (Z p ), with b chosen large enough so that F is a disjoint union of self-similar images. We can describe an iterative process for constructing F. Let K 0 be the convex hull of S in R n. Then F K 0, so F = ( (b 1) v + 1 ) b b F v S ( (b 1) v + 1 ) b b K 0 := K 1 v S We now induct to get compact subsets K n with F = K n We can consider each K n as an approximation of F. In this sense, two p-adic integers are within 1/p n of each other if and only if their images under ψ b are in the same component of K n. As before, this gives the continuity of both ψ b and ψ 1 b when ψ b is injective. 12

13 3.2.1 Examples in R 2 Finally, for the sake of illustration, we describe two nice examples in the Euclidean plane using regular polygons. For these examples, we obtain highly symmetrical fractal subsets of R 2 which are Euclidean models of Z p Take p > 3. Consider the regular (p 1)-gon P in R 2 with vertices v 1,..., v p 1, lying on the boundary of the unit disk in R 2. We define ν : {0,..., p 1} R 2 by ν(0) = 0 and ν(i) = v i for i 0. For example, with p = 5, we take v 1 = (1, 0) and b = 4. Then ψ 4 : Z 5 R 2 is injective and F = ψ 4 (Z 5 ) R 2 is a Euclidean model of Z 5 contained within a square centered at the origin in R 2. Figure 1: The approximating superset K 4 of the fractal subset F = ψ 4 (Z 5 ). With p = 7, we again take v 1 = (1, 0) and b = 4. Then ψ 4 : Z 7 R 2 is injective and F = ψ 4 (Z 7 ) R 2 is a Euclidean model of Z 7 contained within a regular hexagon centered at the origin in R 2. 8 All images were created with care and unwavering determination by the thoughtful and skilled Marc Hester. 13

14 Figure 2: The approximating superset K 4 of the fractal subset F = ψ 4 (Z 7 ). 2. For p 3 consider the regular p-gon P in R 2 with vertices v 0,..., v p 1 lying on the boundary of the unit disk in R 2. We define ν : {0,..., p 1} R 2 by ν(i) = v i and take b = p. For example, with p = 3, we take v 0 = (0, 1). Then ψ 3 : Z 3 R 2 is injective and F = ψ 3 (Z 3 ) R 2 is a Euclidean model of Z 3 contained within a regular triangle centered at the origin in R 2. Figure 3: The approximating superset K 5 of the fractal subset F = ψ 3 (Z 3 ). 14

15 4 Conclusion For the non-number theorist (and perhaps even for some number theorists), the topological space Z p may remain an eccentric example within metric space topology, while for others (including the author) Z p is perhaps the most natural example of a metric space. We have seen how to embed Z p into Euclidean space and thus have described how we may visualize Z p within the familiar context and setting of R n. This perhaps provides additional motivation for introducing and studying the p-adic world. 5 Addendum: Euclidean models of Q p In the preceding section, we described some Euclidean models of Z p. In these cases, we obtained bounded subsets of Euclidean space, reflecting the fact that the p-adic metric is bounded. For this paper, it will suffice to define Q p as the field of fractions of Z p, i.e. Q p is obtained from Z p by inverting all non-zero elements of Z p and is the smallest field (up to isomorphism) containing Z p as a subring 9 Can we describe a similar construction for Q p to obtain fractal subsets of R n which are homeomorphic to Q p? 10 Recall that the group of invertible elements of Z p is Z p = Z p \pz p, so we need only invert p in order to obtain Q p. Elements of Q p have expansions of the form α = a i p i i k with only finitely many terms having negative exponent. In fact, we may write Q p = n 0 p n Z p We simply extend the maps ψ b to Q p in the obvious way: ψ b : a i p i i k i k (b 1) b i+1 ν(a i) Since multiplication by p is a contraction mapping, multiplication by 1 p is a dilation. From the representation of Q p as a union, we can see that ψ b (Q p ) is 9 Alternatively, we could have constructed Q p first as the completion of Q with respect to the p-adic absolute value on Q defined in terms of the p-adic valuation, which we may extend from Z to all of Q. 10 This section was added following my presentation. 15

16 an unbounded fractal subset of R n whose self-similarity structure is identical to that of ψ b (Z p ). We can iteratively build approximations of ψ b (Q p ) from ψ b (Z p ) by arranging dilated copies of ψ b (Z p ) appropriately within R n. References [1] Albert A. Cuoco. Visualizing the p-adic integers. Amer. Math. Monthly, 98(4): , [2] Fernando Q. Gouvêa. p-adic Numbers. An Introduction. Universitext, [3] Svetlana Katok. p-adic Analysis Compared with Real. American Mathematical Society, [4] Neal Koblitz. p-adic Numbers, p-adic Analysis and Zeta-Functions. Springer-Verlag, [5] Alain M. Robert. A Course in p-adic Analysis. Springer-Verlag, [6] W. H. Schikhof. Ultrametric Calculus. An Introduction to p-adic Analysis. Cambridge University Press,

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

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

Topological properties of Z p and Q p and Euclidean models

Topological properties of Z p and Q p and Euclidean models Topological properties of Z p and Q p and Euclidean models Samuel Trautwein, Esther Röder, Giorgio Barozzi November 3, 20 Topology of Q p vs Topology of R Both R and Q p are normed fields and complete

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

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

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

Zair Ibragimov CSUF. Talk at Fullerton College July 14, Geometry of p-adic numbers. Zair Ibragimov CSUF. Valuations on Rational Numbers

Zair Ibragimov CSUF. Talk at Fullerton College July 14, Geometry of p-adic numbers. Zair Ibragimov CSUF. Valuations on Rational Numbers integers Talk at Fullerton College July 14, 2011 integers Let Z = {..., 2, 1, 0, 1, 2,... } denote the integers. Let Q = {a/b : a, b Z and b > 0} denote the rational. We can add and multiply rational :

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

Nonarchimedean Cantor set and string

Nonarchimedean Cantor set and string J fixed point theory appl Online First c 2008 Birkhäuser Verlag Basel/Switzerland DOI 101007/s11784-008-0062-9 Journal of Fixed Point Theory and Applications Nonarchimedean Cantor set and string Michel

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

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

Metric spaces and metrizability

Metric spaces and metrizability 1 Motivation Metric spaces and metrizability By this point in the course, this section should not need much in the way of motivation. From the very beginning, we have talked about R n usual and how relatively

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

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

Math 117: Topology of the Real Numbers

Math 117: Topology of the Real Numbers Math 117: Topology of the Real Numbers John Douglas Moore November 10, 2008 The goal of these notes is to highlight the most important topics presented in Chapter 3 of the text [1] and to provide a few

More information

Liapunov Stability and the ring of P-adic integers

Liapunov Stability and the ring of P-adic integers São Paulo Journal of Mathematical Sciences 2, 1 (2008), 77 84 Liapunov Stability and the ring of P-adic integers Jorge Buescu 1 Dep. Matemática, Fac. Ciências Lisboa, Portugal E-mail address: jbuescu@ptmat.fc.ul.pt

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

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

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9 MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended

More information

After taking the square and expanding, we get x + y 2 = (x + y) (x + y) = x 2 + 2x y + y 2, inequality in analysis, we obtain.

After taking the square and expanding, we get x + y 2 = (x + y) (x + y) = x 2 + 2x y + y 2, inequality in analysis, we obtain. Lecture 1: August 25 Introduction. Topology grew out of certain questions in geometry and analysis about 100 years ago. As Wikipedia puts it, the motivating insight behind topology is that some geometric

More information

MATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES

MATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES MATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES PETE L. CLARK 4. Metric Spaces (no more lulz) Directions: This week, please solve any seven problems. Next week, please solve seven more. Starred parts of

More information

arxiv: v2 [math.ca] 4 Jun 2017

arxiv: v2 [math.ca] 4 Jun 2017 EXCURSIONS ON CANTOR-LIKE SETS ROBERT DIMARTINO AND WILFREDO O. URBINA arxiv:4.70v [math.ca] 4 Jun 07 ABSTRACT. The ternary Cantor set C, constructed by George Cantor in 883, is probably the best known

More information

Connectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ).

Connectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ). Connectedness 1 Motivation Connectedness is the sort of topological property that students love. Its definition is intuitive and easy to understand, and it is a powerful tool in proofs of well-known results.

More information

arxiv:math/ v1 [math.ho] 2 Feb 2005

arxiv:math/ v1 [math.ho] 2 Feb 2005 arxiv:math/0502049v [math.ho] 2 Feb 2005 Looking through newly to the amazing irrationals Pratip Chakraborty University of Kaiserslautern Kaiserslautern, DE 67663 chakrabo@mathematik.uni-kl.de 4/2/2004.

More information

1 Adeles over Q. 1.1 Absolute values

1 Adeles over Q. 1.1 Absolute values 1 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 only if

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

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

Chapter 2 Metric Spaces

Chapter 2 Metric Spaces Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics

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

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

Maths 212: Homework Solutions

Maths 212: Homework Solutions Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then

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

Topological properties

Topological properties CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological

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

Theorems. Theorem 1.11: Greatest-Lower-Bound Property. Theorem 1.20: The Archimedean property of. Theorem 1.21: -th Root of Real Numbers

Theorems. Theorem 1.11: Greatest-Lower-Bound Property. Theorem 1.20: The Archimedean property of. Theorem 1.21: -th Root of Real Numbers Page 1 Theorems Wednesday, May 9, 2018 12:53 AM Theorem 1.11: Greatest-Lower-Bound Property Suppose is an ordered set with the least-upper-bound property Suppose, and is bounded below be the set of lower

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

ZEROS OF SPARSE POLYNOMIALS OVER LOCAL FIELDS OF CHARACTERISTIC p

ZEROS OF SPARSE POLYNOMIALS OVER LOCAL FIELDS OF CHARACTERISTIC p ZEROS OF SPARSE POLYNOMIALS OVER LOCAL FIELDS OF CHARACTERISTIC p BJORN POONEN 1. Statement of results Let K be a field of characteristic p > 0 equipped with a valuation v : K G taking values in an ordered

More information

Metric Spaces and Topology

Metric Spaces and Topology Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies

More information

Numbers 1. 1 Overview. 2 The Integers, Z. John Nachbar Washington University in St. Louis September 22, 2017

Numbers 1. 1 Overview. 2 The Integers, Z. John Nachbar Washington University in St. Louis September 22, 2017 John Nachbar Washington University in St. Louis September 22, 2017 1 Overview. Numbers 1 The Set Theory notes show how to construct the set of natural numbers N out of nothing (more accurately, out of

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

By (a), B ε (x) is a closed subset (which

By (a), B ε (x) is a closed subset (which Solutions to Homework #3. 1. Given a metric space (X, d), a point x X and ε > 0, define B ε (x) = {y X : d(y, x) ε}, called the closed ball of radius ε centered at x. (a) Prove that B ε (x) is always a

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

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

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

B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.

B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X. Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2

More information

Homework in Topology, Spring 2009.

Homework in Topology, Spring 2009. Homework in Topology, Spring 2009. Björn Gustafsson April 29, 2009 1 Generalities To pass the course you should hand in correct and well-written solutions of approximately 10-15 of the problems. For higher

More information

REVIEW OF ESSENTIAL MATH 346 TOPICS

REVIEW OF ESSENTIAL MATH 346 TOPICS REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations

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

CLASS NOTES FOR APRIL 14, 2000

CLASS NOTES FOR APRIL 14, 2000 CLASS NOTES FOR APRIL 14, 2000 Announcement: Section 1.2, Questions 3,5 have been deferred from Assignment 1 to Assignment 2. Section 1.4, Question 5 has been dropped entirely. 1. Review of Wednesday class

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

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

VALUATION THEORY, GENERALIZED IFS ATTRACTORS AND FRACTALS

VALUATION THEORY, GENERALIZED IFS ATTRACTORS AND FRACTALS VALUATION THEORY, GENERALIZED IFS ATTRACTORS AND FRACTALS JAN DOBROWOLSKI AND FRANZ-VIKTOR KUHLMANN Abstract. Using valuation rings and valued fields as examples, we discuss in which ways the notions of

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

MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then

MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever

More information

Solution. 1 Solutions of Homework 1. 2 Homework 2. Sangchul Lee. February 19, Problem 1.2

Solution. 1 Solutions of Homework 1. 2 Homework 2. Sangchul Lee. February 19, Problem 1.2 Solution Sangchul Lee February 19, 2018 1 Solutions of Homework 1 Problem 1.2 Let A and B be nonempty subsets of R + :: {x R : x > 0} which are bounded above. Let us define C = {xy : x A and y B} Show

More information

Mid Term-1 : Practice problems

Mid Term-1 : Practice problems Mid Term-1 : Practice problems These problems are meant only to provide practice; they do not necessarily reflect the difficulty level of the problems in the exam. The actual exam problems are likely to

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

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski

More information

Geometry in the non-archimedean world

Geometry in the non-archimedean world Geometry in the non-archimedean world Annette Werner Strasbourg, June 2014 1 / 29 Geometry in the non-archimedean world Strasbourg, June 2014 Real and complex analysis The fields R and C together with

More information

Course 212: Academic Year Section 1: Metric Spaces

Course 212: Academic Year Section 1: Metric Spaces Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........

More information

ON SPACE-FILLING CURVES AND THE HAHN-MAZURKIEWICZ THEOREM

ON SPACE-FILLING CURVES AND THE HAHN-MAZURKIEWICZ THEOREM ON SPACE-FILLING CURVES AND THE HAHN-MAZURKIEWICZ THEOREM ALEXANDER KUPERS Abstract. These are notes on space-filling curves, looking at a few examples and proving the Hahn-Mazurkiewicz theorem. This theorem

More information

08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms

08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms (February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops

More information

p-adic Continued Fractions

p-adic Continued Fractions p-adic Continued Fractions Matthew Moore May 4, 2006 Abstract Simple continued fractions in R have a single definition and algorithms for calculating them are well known. There also exists a well known

More information

Introductory Analysis I Fall 2014 Homework #5 Solutions

Introductory Analysis I Fall 2014 Homework #5 Solutions Introductory Analysis I Fall 2014 Homework #5 Solutions 6. Let M be a metric space, let C D M. Now we can think of C as a subset of the metric space M or as a subspace of the metric space D (D being a

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

Chapter 2. Metric Spaces. 2.1 Metric Spaces

Chapter 2. Metric Spaces. 2.1 Metric Spaces Chapter 2 Metric Spaces ddddddddddddddddddddddddd ddddddd dd ddd A metric space is a mathematical object in which the distance between two points is meaningful. Metric spaces constitute an important class

More information

4 Countability axioms

4 Countability axioms 4 COUNTABILITY AXIOMS 4 Countability axioms Definition 4.1. Let X be a topological space X is said to be first countable if for any x X, there is a countable basis for the neighborhoods of x. X is said

More information

Axioms of separation

Axioms of separation Axioms of separation These notes discuss the same topic as Sections 31, 32, 33, 34, 35, and also 7, 10 of Munkres book. Some notions (hereditarily normal, perfectly normal, collectionwise normal, monotonically

More information

Topology, Math 581, Fall 2017 last updated: November 24, Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski

Topology, Math 581, Fall 2017 last updated: November 24, Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski Topology, Math 581, Fall 2017 last updated: November 24, 2017 1 Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski Class of August 17: Course and syllabus overview. Topology

More information

3 COUNTABILITY AND CONNECTEDNESS AXIOMS

3 COUNTABILITY AND CONNECTEDNESS AXIOMS 3 COUNTABILITY AND CONNECTEDNESS AXIOMS Definition 3.1 Let X be a topological space. A subset D of X is dense in X iff D = X. X is separable iff it contains a countable dense subset. X satisfies the first

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

MAT1000 ASSIGNMENT 1. a k 3 k. x =

MAT1000 ASSIGNMENT 1. a k 3 k. x = MAT1000 ASSIGNMENT 1 VITALY KUZNETSOV Question 1 (Exercise 2 on page 37). Tne Cantor set C can also be described in terms of ternary expansions. (a) Every number in [0, 1] has a ternary expansion x = a

More information

Do we need Number Theory? Václav Snášel

Do we need Number Theory? Václav Snášel Do we need Number Theory? Václav Snášel What is a number? MCDXIX 664554 0xABCD 01717 010101010111011100001 i i 1 + 1 1! + 1 2! + 1 3! + 1 4! + VŠB-TUO, Ostrava 2014 2 References Neal Koblitz, p-adic Numbers,

More information

01. Review of metric spaces and point-set topology. 1. Euclidean spaces

01. Review of metric spaces and point-set topology. 1. Euclidean spaces (October 3, 017) 01. Review of metric spaces and point-set topology Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 017-18/01

More information

The small ball property in Banach spaces (quantitative results)

The small ball property in Banach spaces (quantitative results) The small ball property in Banach spaces (quantitative results) Ehrhard Behrends Abstract A metric space (M, d) is said to have the small ball property (sbp) if for every ε 0 > 0 there exists a sequence

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

Convergence of a Generalized Midpoint Iteration

Convergence of a Generalized Midpoint Iteration J. Able, D. Bradley, A.S. Moon under the supervision of Dr. Xingping Sun REU Final Presentation July 31st, 2014 Preliminary Words O Rourke s conjecture We begin with a motivating question concerning the

More information

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3 Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,

More information

INDEX. Bolzano-Weierstrass theorem, for sequences, boundary points, bounded functions, 142 bounded sets, 42 43

INDEX. Bolzano-Weierstrass theorem, for sequences, boundary points, bounded functions, 142 bounded sets, 42 43 INDEX Abel s identity, 131 Abel s test, 131 132 Abel s theorem, 463 464 absolute convergence, 113 114 implication of conditional convergence, 114 absolute value, 7 reverse triangle inequality, 9 triangle

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

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

QUOTIENTS OF FIBONACCI NUMBERS

QUOTIENTS OF FIBONACCI NUMBERS QUOTIENTS OF FIBONACCI NUMBERS STEPHAN RAMON GARCIA AND FLORIAN LUCA Abstract. There have been many articles in the Monthly on quotient sets over the years. We take a first step here into the p-adic setting,

More information

Math 426 Homework 4 Due 3 November 2017

Math 426 Homework 4 Due 3 November 2017 Math 46 Homework 4 Due 3 November 017 1. Given a metric space X,d) and two subsets A,B, we define the distance between them, dista,b), as the infimum inf a A, b B da,b). a) Prove that if A is compact and

More information

Hausdorff Measure. Jimmy Briggs and Tim Tyree. December 3, 2016

Hausdorff Measure. Jimmy Briggs and Tim Tyree. December 3, 2016 Hausdorff Measure Jimmy Briggs and Tim Tyree December 3, 2016 1 1 Introduction In this report, we explore the the measurement of arbitrary subsets of the metric space (X, ρ), a topological space X along

More information

1. Continuous Functions between Euclidean spaces

1. Continuous Functions between Euclidean spaces Math 441 Topology Fall 2012 Metric Spaces by John M. Lee This handout should be read between Chapters 1 and 2 of the text. It incorporates material from notes originally prepared by Steve Mitchell and

More information

Existence and Uniqueness

Existence and Uniqueness Chapter 3 Existence and Uniqueness An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect

More information

A VERY BRIEF REVIEW OF MEASURE THEORY

A VERY BRIEF REVIEW OF MEASURE THEORY A VERY BRIEF REVIEW OF MEASURE THEORY A brief philosophical discussion. Measure theory, as much as any branch of mathematics, is an area where it is important to be acquainted with the basic notions and

More information

MA651 Topology. Lecture 9. Compactness 2.

MA651 Topology. Lecture 9. Compactness 2. MA651 Topology. Lecture 9. Compactness 2. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology

More information

Economics 204 Fall 2011 Problem Set 1 Suggested Solutions

Economics 204 Fall 2011 Problem Set 1 Suggested Solutions Economics 204 Fall 2011 Problem Set 1 Suggested Solutions 1. Suppose k is a positive integer. Use induction to prove the following two statements. (a) For all n N 0, the inequality (k 2 + n)! k 2n holds.

More information

MA651 Topology. Lecture 10. Metric Spaces.

MA651 Topology. Lecture 10. Metric Spaces. MA65 Topology. Lecture 0. Metric Spaces. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Linear Algebra and Analysis by Marc Zamansky

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

MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION. Problem 1

MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION. Problem 1 MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION ELEMENTS OF SOLUTION Problem 1 1. Let X be a Hausdorff space and K 1, K 2 disjoint compact subsets of X. Prove that there exist disjoint open sets U 1 and

More information

Rose-Hulman Undergraduate Mathematics Journal

Rose-Hulman Undergraduate Mathematics Journal Rose-Hulman Undergraduate Mathematics Journal Volume 17 Issue 1 Article 5 Reversing A Doodle Bryan A. Curtis Metropolitan State University of Denver Follow this and additional works at: http://scholar.rose-hulman.edu/rhumj

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

(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

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

Topology of Nonarchimedean Analytic Spaces

Topology of Nonarchimedean Analytic Spaces Topology of Nonarchimedean Analytic Spaces AMS Current Events Bulletin Sam Payne January 11, 2013 Complex algebraic geometry Let X C n be an algebraic set, the common solutions of a system of polynomial

More information

p-adic Analysis in Arithmetic Geometry

p-adic Analysis in Arithmetic Geometry p-adic Analysis in Arithmetic Geometry Winter Semester 2015/2016 University of Bayreuth Michael Stoll Contents 1. Introduction 2 2. p-adic numbers 3 3. Newton Polygons 14 4. Multiplicative seminorms and

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

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