Notes on P-Adic Numbers

Total Page:16

File Type:pdf, Size:1020Kb

Notes on P-Adic Numbers P-ADIC NUMBERS JAN-HENDRIK EVERTSE March 2011 Literature: Z.I. Borevich, I.R. Shafarevich, Number Theory, Academic Press, 1966, Chap. 1,4. A. Fr¨ohlich, Local Fields, in: Algebraic Number Theory, edited by J.W.S. Cassels, A. Fr¨ohlich, Academic Press, 1967, Chap. 1. 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. I,III. C. Lech, A note on recurring sequences, Arkiv f¨ormathematik, Band 2, no. 22 (1953), 417{421. L.J. Mordell, Diophantine Equations, Academic Press, 1969, Chap. 23. 1. Absolute values The p-adic absolute value j·jp on Q is defined as follows: if a 2 Q, a 6= 0 then m −m write a = p b=c where b; c are integers not divisible by p and put jajp = p ; further, put j0jp = 0. −7 8 −3 7 −8 3 Example. Let a = −2 3 5 . Then jaj2 = 2 , jaj3 = 3 , jaj5 = 5 , jajp = 1 for p > 7. We give some properties: jabjp = jajpjbjp for a; b 2 Q; ja + bjp 6 max(jajp; jbjp) for a; b 2 Q (ultrametric inequality): Notice that the last property implies that ja + bjp = max(jajp; jbjp) if jajp 6= jbjp: It is common to write the ordinary absolute value jaj = max(a; −a) on Q as jaj1, to call 1 the `infinite prime' and to define MQ := f1g [ fprimesg. 1 2 MASTER COURSE DIOPHANTINE EQUATIONS, SPRING 2011 Then we have the important product formula: Y jajp = 1 for a 2 Q; a 6= 0: p2MQ Absolute values on fields. We define more generally absolute values on fields. Let K be any field. An absolute value on K is a function j·j : K ! R>0 with the following properties: jabj = jaj · jbj for a; b 2 K; ja + bj 6 jaj + jbj for a; b 2 K (triangle inequality); jaj = 0 () a = 0: Note that these properties imply that j1j = 1. The absolute value j · j is called non-trivial if there is a 2 K with jaj 6= f0; 1g. The absolute value j · j is called non-archimedean if the triangle inequality can be replaced by the stronger ultrametric inequality ja + bj 6 max(jaj; jbj) for a; b 2 K: An absolute value not satisfying the ultrametric inequality is called archimedean. If K is a field with absolute value j · j and L an extension of K, then an extension or continuation of j·j to L is an absolute value on L whose restriction to K is j · j. Examples. 1) The ordinary absolute value j · j on Q is archimedean, while the p-adic absolute values are all non-archimedean. 2) Let K be any field, and K(t) the field of rational functions of K. For a polynomial f 2 K[t] define jfj = 0 if f = 0 and jfj = edeg f if f 6= 0. Further, for a rational function f=g with f; g 2 K[t] define jf=gj = jfj=jgj. Verify that this defines a non-archimedean absolute value on K(t). Two absolute values j · j1; j · j2 on K are called equivalent if there is α > 0 α such that jxj2 = jxj1 for all x 2 K. We state without proof the following result: Theorem 1.1 (Ostrowski). Every non-trivial absolute value on Q is equivalent to either the ordinary absolute value or a p-adic absolute value for some prime number p. P-ADIC NUMBERS 3 Valuations. In algebra and number theory, one quite often deals with val- uations instead of absolute values. A valuation on a field K is a function v : K ! R [ f1g such that for some constant c > 1, c−v(·) defines a non- archimedean absolute value on K. That is, v(x) = 1 () x = 0; v(xy) = v(x) + v(y) for x; y 2 K; v(x + y) > min(v(x); v(y)) for x; y 2 K. The valuation is called non-trivial if there is a 2 K∗ with v(a) 6= 0. The set v(K∗) is an additive subgroup of R. The valuation v is called discrete if v(K∗) is a discrete subgroup of R. A normalized discrete valuation is one for which v(K∗) = Z. 2. Completions An absolute value preserving isomorphism between two fields K1;K2 with absolute values j · j1, j · j2, respectively, is an isomorphism ' : K1 ! K2 such that j'(x)j2 = jxj1 for x 2 K1. 1 Let K be a field, j · j a non-trivial absolute value on K, and fakgk=0 a sequence in K. 1 We say that fakgk=0 converges to α with respect to j · j if limk!1 jak − αj = 0. 1 Further, fakgk=0 is called a Cauchy sequence with respect to j · j if limm;n!1 jam − anj = 0. Notice that any convergent sequence is a Cauchy sequence. We say that K is complete with respect to j · j if every Cauchy sequence w.r.t. j · j in K converges to a limit in K. For instance, R and C are complete w.r.t. the ordinary absolute value. By mimicking the construction of R from Q, one can show that every field K with an absolute value can be extended to an essentially unique field Ke, such that Ke is complete and every element of Ke is the limit of a Cauchy sequence from K. Theorem 2.1. Let K be a field with absolute value j · j. There is an up to absolute value preserving isomorphism unique extension field Ke of K, called the completion of K, having the following properties: 4 MASTER COURSE DIOPHANTINE EQUATIONS, SPRING 2011 (i) j · j can be continued to an absolute value on Ke, also denoted j · j, such that Ke is complete w.r.t. j · j; (ii) K is dense in Ke, i.e., every element of Ke is the limit of a sequence from K. Proof. We give a sketch. Cauchy sequences, limits, etc. are all with respect to j · j. The set of Cauchy sequences in K with respect to j·j is closed under termwise addition and multiplication fang + fbng := fan + bng, fang · fbng := fan · bng. With these operations they form a ring, which we denote by R. It is not difficult to verify that the sequences fang such that an ! 0 with respect to j · j form a maximal ideal in R, which we denote by M. Thus, the quotient R=M is a field, which is our completion Ke. We define the absolute value jαj of α 2 Ke by choosing a representative fang of α, and putting jαj := limn!1 janj, where now the limit is with respect to the ordinary absolute value on R. It is not difficult to verify that this is well-defined, that is, the limit exists and is independent of the choice of the representative fang. We may view K as a subfield of Ke by identifying a 2 K with the element of Ke represented by the constant Cauchy sequence fag. In this manner, the absolute value on Ke constructed above extends that of K, and moreover, every element of Ke is a limit of a sequence from K. So K is dense in Ke. One shows that Ke is complete, that is, any Cauchy sequence fang in Ke has a limit in Ke, by taking very good approximations bn 2 K of an and then taking the limit of the bn. Finally, if K0 is another complete field with absolute value extending that on K such that K is dense in K0 one obtains an isomorphism from Ke to K0 as follows: Take α 2 Ke. Choose a sequence fakg in K converging to α; this is 0 necessarily a Cauchy sequence. Then map α to the limit of fakg in K . 2 Corollary 2.2. Assume that j · j is a non-archimedean absolute value on K. Then the extension of j · j to Ke is also non-archimedean. Proof. Let a; b 2 Ke. Choose sequences fakg, fbkg in K that converge to a, b, respectively. Then taking the limit of jak + bkj 6 max(jakj; jbkj) gives ja + bj 6 max(jaj; jbj). 2 P-ADIC NUMBERS 5 Ostrowski proved that any field complete with respect to an archimedean absolute value is isomorphic to R or C. As a consequence, any field that can be endowed with an archimedean absolute value is isomorphic to a subfield of C. On the other hand, there is a much larger variety of fields with a non- archimedean absolute value. It is possible to define notions such as convergence, continuity, differentia- bility, etc. for complete fields with a non-archimedean absolute value similarly as for R or C, and this leads to non-archimedean analysis. One of the striking features of non-archimedean analysis is the following very easy criterion for convergence of series. Lemma 2.3. Let K be a field complete w.r.t. a non-archimedean absolute 1 P1 value j · j. Let fakgk=0 be a sequence in K. Then k=0 ak converges in K if and only if limk!1 ak = 0. P1 Proof. Suppose that α := k=0 ak converges. Then n n−1 X X an = ak − ak ! α − α = 0: k=0 k=0 Pn Conversely, suppose that ak ! 0 as k ! 1. Let αn := k=0 ak. Then for any integers m; n with 0 < m < n we have n X jαn − αmj = j akj 6 max(jam+1j;:::; janj) ! 0 as m; n ! 1 : k=m+1 So the partial sums αn form a Cauchy sequence, hence must converge to a limit in K.
Recommended publications
  • Formal Power Series - Wikipedia, the Free Encyclopedia
    Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series
    [Show full text]
  • The Axiom of Choice, Zorn's Lemma, and the Well
    THE AXIOM OF CHOICE, ZORN'S LEMMA, AND THE WELL ORDERING PRINCIPLE The Axiom of Choice is a foundational statement of set theory: Given any collection fSi : i 2 Ig of nonempty sets, there exists a choice function [ f : I −! Si; f(i) 2 Si for all i 2 I: i In the 1930's, Kurt G¨odelproved that the Axiom of Choice is consistent (in the Zermelo-Frankel first-order axiomatization) with the other axioms of set theory. In the 1960's, Paul Cohen proved that the Axiom of Choice is independent of the other axioms. Closely following Van der Waerden, this writeup explains how the Axiom of Choice implies two other statements, Zorn's Lemma and the Well Ordering Prin- ciple. In fact, all three statements are equivalent, as is a fourth statement called the Hausdorff Maximality Principle. 1. Partial Order Let S be any set, and let P(S) be its power set, the set whose elements are the subsets of S, P(S) = fS : S is a subset of Sg: The proper subset relation \(" in P(S) satisfies two conditions: • (Partial Trichotomy) For any S; T 2 P(S), at most one of the conditions S ( T;S = T;T ( S holds, and possibly none of them holds. • (Transitivity) For any S; T; U 2 P(S), if S ( T and T ( U then also S ( U. The two conditions are the motivating example of a partial order. Definition 1.1. Let T be a set. Let ≺ be a relation that may stand between some pairs of elements of T .
    [Show full text]
  • An Introduction to Skolem's P-Adic Method for Solving Diophantine
    An introduction to Skolem's p-adic method for solving Diophantine equations Josha Box July 3, 2014 Bachelor thesis Supervisor: dr. Sander Dahmen Korteweg-de Vries Instituut voor Wiskunde Faculteit der Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam Abstract In this thesis, an introduction to Skolem's p-adic method for solving Diophantine equa- tions is given. The main theorems that are proven give explicit algorithms for computing bounds for the amount of integer solutions of special Diophantine equations of the kind f(x; y) = 1, where f 2 Z[x; y] is an irreducible form of degree 3 or 4 such that the ring of integers of the associated number field has one fundamental unit. In the first chapter, an introduction to algebraic number theory is presented, which includes Minkovski's theorem and Dirichlet's unit theorem. An introduction to p-adic numbers is given in the second chapter, ending with the proof of the p-adic Weierstrass preparation theorem. The theory of the first two chapters is then used to apply Skolem's method in Chapter 3. Title: An introduction to Skolem's p-adic method for solving Diophantine equations Author: Josha Box, [email protected], 10206140 Supervisor: dr. Sander Dahmen Second grader: Prof. dr. Jan de Boer Date: July 3, 2014 Korteweg-de Vries Instituut voor Wiskunde Universiteit van Amsterdam Science Park 904, 1098 XH Amsterdam http://www.science.uva.nl/math 3 Contents Introduction 6 Motivations . .7 1 Number theory 8 1.1 Prerequisite knowledge . .8 1.2 Algebraic numbers . 10 1.3 Unique factorization . 15 1.4 A geometrical approach to number theory .
    [Show full text]
  • Section V.1. Field Extensions
    V.1. Field Extensions 1 Section V.1. Field Extensions Note. In this section, we define extension fields, algebraic extensions, and tran- scendental extensions. We treat an extension field F as a vector space over the subfield K. This requires a brief review of the material in Sections IV.1 and IV.2 (though if time is limited, we may try to skip these sections). We start with the definition of vector space from Section IV.1. Definition IV.1.1. Let R be a ring. A (left) R-module is an additive abelian group A together with a function mapping R A A (the image of (r, a) being × → denoted ra) such that for all r, a R and a,b A: ∈ ∈ (i) r(a + b)= ra + rb; (ii) (r + s)a = ra + sa; (iii) r(sa)=(rs)a. If R has an identity 1R and (iv) 1Ra = a for all a A, ∈ then A is a unitary R-module. If R is a division ring, then a unitary R-module is called a (left) vector space. Note. A right R-module and vector space is similarly defined using a function mapping A R A. × → Definition V.1.1. A field F is an extension field of field K provided that K is a subfield of F . V.1. Field Extensions 2 Note. With R = K (the ring [or field] of “scalars”) and A = F (the additive abelian group of “vectors”), we see that F is a vector space over K. Definition. Let field F be an extension field of field K.
    [Show full text]
  • Introduction to the P-Adic Space
    Introduction to the p-adic Space Joel P. Abraham October 25, 2017 1 Introduction From a young age, students learn fundamental operations with the real numbers: addition, subtraction, multiplication, division, etc. We intuitively understand that distance under the reals as the absolute value function, even though we never have had a fundamental introduction to set theory or number theory. As such, I found this topic to be extremely captivating, since it almost entirely reverses the intuitive notion of distance in the reals. Such a small modification to the definition of distance gives rise to a completely different set of numbers under which the typical axioms are false. Furthermore, this is particularly interesting since the p-adic system commonly appears in natural phenomena. Due to the versatility of this metric space, and its interesting uniqueness, I find the p-adic space a truly fascinating development in algebraic number theory, and thus chose to explore it further. 2 The p-adic Metric Space 2.1 p-adic Norm Definition 2.1. A metric space is a set with a distance function or metric, d(p, q) defined over the elements of the set and mapping them to a real number, satisfying: arXiv:1710.08835v1 [math.HO] 22 Oct 2017 (a) d(p, q) > 0 if p = q 6 (b) d(p,p)=0 (c) d(p, q)= d(q,p) (d) d(p, q) d(p, r)+ d(r, q) ≤ The real numbers clearly form a metric space with the absolute value function as the norm such that for p, q R ∈ d(p, q)= p q .
    [Show full text]
  • Local-Global Methods in Algebraic Number Theory
    LOCAL-GLOBAL METHODS IN ALGEBRAIC NUMBER THEORY ZACHARY KIRSCHE Abstract. This paper seeks to develop the key ideas behind some local-global methods in algebraic number theory. To this end, we first develop the theory of local fields associated to an algebraic number field. We then describe the Hilbert reciprocity law and show how it can be used to develop a proof of the classical Hasse-Minkowski theorem about quadratic forms over algebraic number fields. We also discuss the ramification theory of places and develop the theory of quaternion algebras to show how local-global methods can also be applied in this case. Contents 1. Local fields 1 1.1. Absolute values and completions 2 1.2. Classifying absolute values 3 1.3. Global fields 4 2. The p-adic numbers 5 2.1. The Chevalley-Warning theorem 5 2.2. The p-adic integers 6 2.3. Hensel's lemma 7 3. The Hasse-Minkowski theorem 8 3.1. The Hilbert symbol 8 3.2. The Hasse-Minkowski theorem 9 3.3. Applications and further results 9 4. Other local-global principles 10 4.1. The ramification theory of places 10 4.2. Quaternion algebras 12 Acknowledgments 13 References 13 1. Local fields In this section, we will develop the theory of local fields. We will first introduce local fields in the special case of algebraic number fields. This special case will be the main focus of the remainder of the paper, though at the end of this section we will include some remarks about more general global fields and connections to algebraic geometry.
    [Show full text]
  • Algebraic Number Theory
    Algebraic Number Theory William B. Hart Warwick Mathematics Institute Abstract. We give a short introduction to algebraic number theory. Algebraic number theory is the study of extension fields Q(α1; α2; : : : ; αn) of the rational numbers, known as algebraic number fields (sometimes number fields for short), in which each of the adjoined complex numbers αi is algebraic, i.e. the root of a polynomial with rational coefficients. Throughout this set of notes we use the notation Z[α1; α2; : : : ; αn] to denote the ring generated by the values αi. It is the smallest ring containing the integers Z and each of the αi. It can be described as the ring of all polynomial expressions in the αi with integer coefficients, i.e. the ring of all expressions built up from elements of Z and the complex numbers αi by finitely many applications of the arithmetic operations of addition and multiplication. The notation Q(α1; α2; : : : ; αn) denotes the field of all quotients of elements of Z[α1; α2; : : : ; αn] with nonzero denominator, i.e. the field of rational functions in the αi, with rational coefficients. It is the smallest field containing the rational numbers Q and all of the αi. It can be thought of as the field of all expressions built up from elements of Z and the numbers αi by finitely many applications of the arithmetic operations of addition, multiplication and division (excepting of course, divide by zero). 1 Algebraic numbers and integers A number α 2 C is called algebraic if it is the root of a monic polynomial n n−1 n−2 f(x) = x + an−1x + an−2x + ::: + a1x + a0 = 0 with rational coefficients ai.
    [Show full text]
  • Math 121. Uniqueness of Algebraic Closure Let K Be a Field, and K/K A
    Math 121. Uniqueness of algebraic closure Let k be a field, and k=k a choice of algebraic closure. As a first step in the direction of proving that k is \unique up to (non-unique) isomorphism", we prove: Lemma 0.1. Let L=k be an algebraic extension, and L0=L another algebraic extension. There is a k-embedding i : L,! k, and once i is picked there exists a k-embedding L0 ,! k extending i. Proof. Since an embedding i : L,! k realizes the algebraically closed k as an algebraic extension of L (and hence as an algebraic closure of L), by renaming the base field as L it suffices to just prove the first part: any algebraic extension admits an embedding into a specified algebraic closure. Define Σ to be the set of pairs (k0; i) where k0 ⊆ L is an intermediate extension over k and 0 i : k ,! k is a k-embedding. Using the inclusion i0 : k ,! k that comes along with the data of how k is realized as an algebraic closure of k, we see that (k; i0) 2 Σ, so Σ is non-empty. We wish to apply Zorn's Lemma, where we define a partial ordering on Σ by the condition that (k0; i0) ≤ (k00; i00) if 0 00 00 0 k ⊆ k inside of L and i jk0 = i . It is a simple exercise in gluing set maps to see that the hypothesis of Zorn's Lemma is satisfied, so there exists a maximal element (K; i) 2 Σ. We just have to show K = L.
    [Show full text]
  • Geometry, Combinatorial Designs and Cryptology Fourth Pythagorean Conference
    Geometry, Combinatorial Designs and Cryptology Fourth Pythagorean Conference Sunday 30 May to Friday 4 June 2010 Index of Talks and Abstracts Main talks 1. Simeon Ball, On subsets of a finite vector space in which every subset of basis size is a basis 2. Simon Blackburn, Honeycomb arrays 3. G`abor Korchm`aros, Curves over finite fields, an approach from finite geometry 4. Cheryl Praeger, Basic pregeometries 5. Bernhard Schmidt, Finiteness of circulant weighing matrices of fixed weight 6. Douglas Stinson, Multicollision attacks on iterated hash functions Short talks 1. Mari´en Abreu, Adjacency matrices of polarity graphs and of other C4–free graphs of large size 2. Marco Buratti, Combinatorial designs via factorization of a group into subsets 3. Mike Burmester, Lightweight cryptographic mechanisms based on pseudorandom number generators 4. Philippe Cara, Loops, neardomains, nearfields and sets of permutations 5. Ilaria Cardinali, On the structure of Weyl modules for the symplectic group 6. Bill Cherowitzo, Parallelisms of quadrics 7. Jan De Beule, Large maximal partial ovoids of Q−(5, q) 8. Bart De Bruyn, On extensions of hyperplanes of dual polar spaces 1 9. Frank De Clerck, Intriguing sets of partial quadrangles 10. Alice Devillers, Symmetry properties of subdivision graphs 11. Dalibor Froncek, Decompositions of complete bipartite graphs into generalized prisms 12. Stelios Georgiou, Self-dual codes from circulant matrices 13. Robert Gilman, Cryptology of infinite groups 14. Otokar Groˇsek, The number of associative triples in a quasigroup 15. Christoph Hering, Latin squares, homologies and Euler’s conjecture 16. Leanne Holder, Bilinear star flocks of arbitrary cones 17. Robert Jajcay, On the geometry of cages 18.
    [Show full text]
  • Pseudo Real Closed Field, Pseudo P-Adically Closed Fields and NTP2
    Pseudo real closed fields, pseudo p-adically closed fields and NTP2 Samaria Montenegro∗ Université Paris Diderot-Paris 7† Abstract The main result of this paper is a positive answer to the Conjecture 5.1 of [15] by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC field, then T h(M) is NTP2 if and only if M is bounded. In the case of PpC fields, we prove that if M is a bounded PpC field, then T h(M) is NTP2. We also generalize this result to obtain that, if M is a bounded PRC or PpC field with exactly n orders or p-adic valuations respectively, then T h(M) is strong of burden n. This also allows us to explicitly compute the burden of types, and to describe forking. Keywords: Model theory, ordered fields, p-adic valuation, real closed fields, p-adically closed fields, PRC, PpC, NIP, NTP2. Mathematics Subject Classification: Primary 03C45, 03C60; Secondary 03C64, 12L12. Contents 1 Introduction 2 2 Preliminaries on pseudo real closed fields 4 2.1 Orderedfields .................................... 5 2.2 Pseudorealclosedfields . .. .. .... 5 2.3 The theory of PRC fields with n orderings ..................... 6 arXiv:1411.7654v2 [math.LO] 27 Sep 2016 3 Bounded pseudo real closed fields 7 3.1 Density theorem for PRC bounded fields . ...... 8 3.1.1 Density theorem for one variable definable sets . ......... 9 3.1.2 Density theorem for several variable definable sets. ........... 12 3.2 Amalgamation theorems for PRC bounded fields . ........ 14 ∗[email protected]; present address: Universidad de los Andes †Partially supported by ValCoMo (ANR-13-BS01-0006) and the Universidad de Costa Rica.
    [Show full text]
  • Ring (Mathematics) 1 Ring (Mathematics)
    Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right.
    [Show full text]
  • Real Closed Fields
    University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 1968 Real closed fields Yean-mei Wang Chou The University of Montana Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Chou, Yean-mei Wang, "Real closed fields" (1968). Graduate Student Theses, Dissertations, & Professional Papers. 8192. https://scholarworks.umt.edu/etd/8192 This Thesis is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. EEAL CLOSED FIELDS By Yean-mei Wang Chou B.A., National Taiwan University, l96l B.A., University of Oregon, 19^5 Presented in partial fulfillment of the requirements for the degree of Master of Arts UNIVERSITY OF MONTANA 1968 Approved by: Chairman, Board of Examiners raduate School Date Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: EP38993 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI OwMTtation PVblmhing UMI EP38993 Published by ProQuest LLC (2013). Copyright in the Dissertation held by the Author.
    [Show full text]