DOCSLIB.ORG

MA3A6 Algebraic Number Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
MA3A6 Algebraic Number Theory MA3A6 Algebraic Number Theory David Loeffler Term 2, 2014–15 Chapter 0 Introduction Lecture 1 0.1 What is this module about? This is a module about algebraic number fields. An algebraic number field is a special kind of field, which contains the rational numbers Q, but is a little bit bigger. We’ll give a formal definition soon enough, but a good example to bear in mind is the Gaussian field Q(i) = fa + bi : a, b 2 Qg, which comes with its subring of Gaussian integers Z[i] = fa + bi : a, b 2 Zg. Exercise. Why is the Gaussian field a field? (Most of the axioms are straightforward, but why is it closed under inverses?) In Algebra 2 you saw that Z[i] was a unique factorization domain, and you used this to show that any prime number p = 1 mod 4 could be written as the sum of two squares, p = x2 + y2. So rings like Z[i] have some interesting structure; and they tell us new things about Z. 0.2 Logistics • There will be 4 problem sheets, which will be distributed as we go along. These count for 15% of your grade. The deadlines will be – Sheet 1: distributed Thursday, week 2; deadline 3pm Monday, week 4. – Sheet 2: distributed Thursday, week 4; deadline 3pm Monday, week 6. – Sheet 3: distributed Thursday, week 6; deadline 3pm Monday, week 8. – Sheet 4: distributed Thursday, week 8; deadline 3pm Monday, week 10. • Weekly office hour: Tuesdays 13.30–14.30, Zeeman B1.25. • Support classes with Heline Deconinck: Fridays 11–12, MS.04, from week 2 onwards. 1 • Books: see list on Undergraduate Handbook page. The main reference is Stewart & Tall, which is also probably the friendliest of the books on the list; Swinnerton-Dyer’s book is harder going, but was the book which inspired me to become a number theorist. • Most of you have done Galois theory, and about half of you are doing Commutative Algebra. 2 Chapter 1 Algebraic number fields 1.1 Extensions of fields Notation 1.1.1. Let K and L be fields. If K is a subfield of L, we say L is a field extension of K, and we write L j K. For instance, C j Q is a field extension, as is C j R. Definition 1.1.2. Let L j K be a field extension, and let a 2 L. We say a is algebraic over K if there exists a nonzero polynomial g 2 K[X] such that g(a) = 0. Example 1.1.3. In the extension C j R, the element ip is algebraic over R (it’s a root of X2 + p2). However, it is not algebraic over Q. Proposition 1.1.4. Let a be algebraic over K. Then there is a unique polynomial f 2 K[X] such that f (a) = 0 and f is irreducible and monic (its leading coefficient is 1). We call this the minimal polynomial of f over K. Proof. Recall from Algebra 2 the concept of an ideal and a principal ideal. The set I ⊂ K[X] of polynomials g such that g(a) = 0 is an ideal of K[X]; the ring K[X] is a Euclidean domain, so every ideal of this ring is principal, i.e. consists of the multiples of some polynomial f (which we can assume is monic, by multiplying it by an element of K× if necessary). To see that f is irreducible, we suppose that we can write f = gh. Then g(a)h(a) = 0; since L is a field, we must have either g(a) = 0 or h(a) = 0, and thus at least one of g and h is in I. So f divides one of g and h, WLOG g. Since g also divides f , we have deg(g) = deg( f ) and hence h is constant. Thus f is irreducible. Remark. For Commutative Algebra students: a slightly posher way of stating the last part is that I is the kernel of the homomorphism K[X] ! L g 7! g(a). L is an integral domain (being a field); the kernel of a homomorphism to an integral domain is a prime ideal; and a generator of a principal prime ideal is a prime element, and hence must be irreducible. Definition 1.1.5. Let L j K be an extension. We say L j K is algebraic if every a 2 L is algebraic over K. We say L j K is finite if L has finite dimension as a K-vector space. Example 1.1.6. The extension C j R is finite (of degree 2), since f1, ig is a basis of C over R. It is also algebraic, because every a + bi 2 C satisfies the polynomial (X − a)2 + b2 = X2 − 2aX + (a2 + b2) 2 R[X]. Notation 1.1.7. If L j K is finite, we define the degree [L : K] to be the dimension of L as a K-vector space. 3 p Example 1.1.8. Let a = i + 2 2 C. But a is also algebraic over Q: we have p p a − 2 = i ) a2 − 2 2a + 2 = −1 p ) a2 + 3 = 2 2a ) (a2 + 3)2 = 8a2 ) a4 − 2a2 + 9 = 0. We’ll see later that X4 − 2X2 + 9 is irreducible in Q[X], so it is the minimal polynomial of a over Q. Lecture p 2 On the other hand, the minimal polynomial of a over R is X2 − 2 2X + 3, by the previous example. This shows that the minimal polynomial of a over K really depends on which K we use! Remark. I forgot to point out in the last lecture that in Proposition 1.1.4, the minimal polynomial f of a over K has the property that any polynomial g 2 K[X] such that g(a) = 0 is necessarily a multiple of f . This is clear from the proof. We’ll use this fact a lot, so make sure it’s in your notes! Proposition 1.1.9. Let L j K be a field extension. An element a 2 L is algebraic over K if and only if there exists a finite extension of K inside L which contains a. (In particular, any finite extension is algebraic, and any algebraic extension is a union of finite extensions. There are algebraic extensions which aren’t finite, as we’ll see later.) Proof. Firstly, let’s prove the “if” part. It suffices to show that if L j K is a finite extension and a 2 L, then a is algebraic. Suppose [L : K] = d < ¥. Then the powers 1, a, a2, ... , ad are d + 1 elements of a d-dimensional vector space over K, so they must be linearly dependent: that is, we can find elements c0, ... , cd of K, not all zero, such that d c0 + c1a + ··· + cda = 0. i Thus a satisfies the non-zero polynomial g(X) = ∑ ciX 2 K[X] of degree ≤ d. Thus a is algebraic over K. The “only if” part is a little harder. Let f be the minimal polynomial of a over K, and d its degree. We will show that the K-subspace M of L spanned by the powers of a is d-dimensional over L, with basis S = f1, ... , ad−1g, and is a subfield of L. Since S is a finite set, this shows that M is a finite field extension of K inside L which contains a. Claim 1: M is a subring of L. N By definition, M is exactly the elements of L which are of the form a0 + a1a + ··· + aNa for some a0, ... , aN 2 K; that is, L is the image of the ring homomorphism K[X] ! L given by mapping g to g(a). But the image of a ring homomorphism is always a subring (Algebra 2). Claim 2: M is spanned by S. By the division algorithm for polynomials, for each g 2 K[X] we can write g(X) = a(X) f (X) + b(X) where a, b 2 K[X] and deg(b) ≤ d − 1. But this implies that g(a) = a(a) f (a) + b(a) = b(a), since f (a) = 0. As b has degree ≤ d − 1, b(a) is a K-linear combination of the elements of S. Claim 3: M is closed under taking inverses of nonzero elements. This is the most difficult bit! There are many possible proofs, but here’s one. We know by this stage that M is finite-dimensional over K. Let x 2 M be non-zero, and consider the map mx : M ! M given by mx(y) = xy. (This is called the “multiplication-by-x map”). I claim that mx is injective. If not, there would be some nonzero y such that xy = 0; but this equality takes place inside L, which is a field, so either x = 0 or y = 0, which is a contradiction. 4 By the rank–nullity theorem, it follows that mx is surjective. In particular, 1 2 image(mx), which shows that 1/x 2 M. This concludes the proof that M is a field. As a by-product of the proof of the “only if” part, we get two interesting pieces of information. Corollary 1.1.10. (i) An element a 2 L is algebraic over K if and only if the powers of a span a finite-dimensional K-subspace of L. (ii) If a is algebraic over L, then there is a unique smallest extension of K in L which contains a, namely the K-subspace spanned by the powers of a; and this has a K-basis 1, a, ..
Load more

Recommended publications
  • The Class Number One Problem for Imaginary Quadratic Fields
    MODULAR CURVES AND THE CLASS NUMBER ONE PROBLEM JEREMY BOOHER Gauss found 9 imaginary quadratic fields with class number one, and in the early 19th century conjectured he had found all of them. It turns out he was correct, but it took until the mid 20th century to prove this. Theorem 1. Let K be an imaginary quadratic field whose ring of integers has class number one. Then K is one of p p p p p p p p Q(i); Q( −2); Q( −3); Q( −7); Q( −11); Q( −19); Q( −43); Q( −67); Q( −163): There are several approaches. Heegner [9] gave a proof in 1952 using the theory of modular functions and complex multiplication. It was dismissed since there were gaps in Heegner's paper and the work of Weber [18] on which it was based. In 1967 Stark gave a correct proof [16], and then noticed that Heegner's proof was essentially correct and in fact equiv- alent to his own. Also in 1967, Baker gave a proof using lower bounds for linear forms in logarithms [1]. Later, Serre [14] gave a new approach based on modular curve, reducing the class number + one problem to finding special points on the modular curve Xns(n). For certain values of n, it is feasible to find all of these points. He remarks that when \N = 24 An elliptic curve is obtained. This is the level considered in effect by Heegner." Serre says nothing more, and later writers only repeat this comment. This essay will present Heegner's argument, as modernized in Cox [7], then explain Serre's strategy.
    [Show full text]
  • Introduction to Class Field Theory and Primes of the Form X + Ny
    Introduction to Class Field Theory and Primes of the Form x2 + ny2 Che Li October 3, 2018 Abstract This paper introduces the basic theorems of class field theory, based on an exposition of some fundamental ideas in algebraic number theory (prime decomposition of ideals, ramification theory, Hilbert class field, and generalized ideal class group), to answer the question of which primes can be expressed in the form x2 + ny2 for integers x and y, for a given n. Contents 1 Number Fields1 1.1 Prime Decomposition of Ideals..........................1 1.2 Basic Ramification Theory.............................3 2 Quadratic Fields6 3 Class Field Theory7 3.1 Hilbert Class Field.................................7 3.2 p = x2 + ny2 for infinitely n’s (1)........................8 3.3 Example: p = x2 + 5y2 .............................. 11 3.4 Orders in Imaginary Quadratic Fields...................... 13 3.5 Theorems of Class Field Theory.......................... 16 3.6 p = x2 + ny2 for infinitely many n’s (2)..................... 18 3.7 Example: p = x2 + 27y2 .............................. 20 1 Number Fields 1.1 Prime Decomposition of Ideals We will review some basic facts from algebraic number theory, including Dedekind Domain, unique factorization of ideals, and ramification theory. To begin, we define an algebraic number field (or, simply, a number field) to be a finite field extension K of Q. The set of algebraic integers in K form a ring OK , which we call the ring of integers, i.e., OK is the set of all α 2 K which are roots of a monic integer polynomial. In general, OK is not a UFD but a Dedekind domain.
    [Show full text]
  • APPLICATIONS of GALOIS THEORY 1. Finite Fields Let F Be a Finite Field
    CHAPTER IX APPLICATIONS OF GALOIS THEORY 1. Finite Fields Let F be a finite field. It is necessarily of nonzero characteristic p and its prime field is the field with p r elements Fp.SinceFis a vector space over Fp,itmusthaveq=p elements where r =[F :Fp]. More generally, if E ⊇ F are both finite, then E has qd elements where d =[E:F]. As we mentioned earlier, the multiplicative group F ∗ of F is cyclic (because it is a finite subgroup of the multiplicative group of a field), and clearly its order is q − 1. Hence each non-zero element of F is a root of the polynomial Xq−1 − 1. Since 0 is the only root of the polynomial X, it follows that the q elements of F are roots of the polynomial Xq − X = X(Xq−1 − 1). Hence, that polynomial is separable and F consists of the set of its roots. (You can also see that it must be separable by finding its derivative which is −1.) We q may now conclude that the finite field F is the splitting field over Fp of the separable polynomial X − X where q = |F |. In particular, it is unique up to isomorphism. We have proved the first part of the following result. Proposition. Let p be a prime. For each q = pr, there is a unique (up to isomorphism) finite field F with |F | = q. Proof. We have already proved the uniqueness. Suppose q = pr, and consider the polynomial Xq − X ∈ Fp[X]. As mentioned above Df(X)=−1sof(X) cannot have any repeated roots in any extension, i.e.
    [Show full text]
  • On Free Quasigroups and Quasigroup Representations Stefanie Grace Wang Iowa State University
    Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2017 On free quasigroups and quasigroup representations Stefanie Grace Wang Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Mathematics Commons Recommended Citation Wang, Stefanie Grace, "On free quasigroups and quasigroup representations" (2017). Graduate Theses and Dissertations. 16298. https://lib.dr.iastate.edu/etd/16298 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. On free quasigroups and quasigroup representations by Stefanie Grace Wang A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Jonathan D.H. Smith, Major Professor Jonas Hartwig Justin Peters Yiu Tung Poon Paul Sacks The student author and the program of study committee are solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred. Iowa State University Ames, Iowa 2017 Copyright c Stefanie Grace Wang, 2017. All rights reserved. ii DEDICATION I would like to dedicate this dissertation to the Integral Liberal Arts Program. The Program changed my life, and I am forever grateful. It is as Aristotle said, \All men by nature desire to know." And Montaigne was certainly correct as well when he said, \There is a plague on Man: his opinion that he knows something." iii TABLE OF CONTENTS LIST OF TABLES .
    [Show full text]
  • A Note on Presentation of General Linear Groups Over a Finite Field
    Southeast Asian Bulletin of Mathematics (2019) 43: 217–224 Southeast Asian Bulletin of Mathematics c SEAMS. 2019 A Note on Presentation of General Linear Groups over a Finite Field Swati Maheshwari and R. K. Sharma Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India Email: [email protected]; [email protected] Received 22 September 2016 Accepted 20 June 2018 Communicated by J.M.P. Balmaceda AMS Mathematics Subject Classification(2000): 20F05, 16U60, 20H25 Abstract. In this article we have given Lie regular generators of linear group GL(2, Fq), n where Fq is a finite field with q = p elements. Using these generators we have obtained presentations of the linear groups GL(2, F2n ) and GL(2, Fpn ) for each positive integer n. Keywords: Lie regular units; General linear group; Presentation of a group; Finite field. 1. Introduction Suppose F is a finite field and GL(n, F) is the general linear the group of n × n invertible matrices and SL(n, F) is special linear group of n × n matrices with determinant 1. We know that GL(n, F) can be written as a semidirect product, GL(n, F)= SL(n, F) oF∗, where F∗ denotes the multiplicative group of F. Let H and K be two groups having presentations H = hX | Ri and K = hY | Si, then a presentation of semidirect product of H and K is given by, −1 H oη K = hX, Y | R,S,xyx = η(y)(x) ∀x ∈ X,y ∈ Y i, where η : K → Aut(H) is a group homomorphism. Now we summarize some literature survey related to the presentation of groups.
    [Show full text]
  • The General Linear Group
    18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices.
    [Show full text]
  • 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]
  • 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]
  • 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]