THE CONDUCTOR IDEAL 1. Introduction Let O Be an Order in The

Total Page:16

File Type:pdf, Size:1020Kb

THE CONDUCTOR IDEAL 1. Introduction Let O Be an Order in The THE CONDUCTOR IDEAL KEITH CONRAD 1. Introduction Let O be an order in the number field K. When O 6= OK , O is Noetherian and one- dimensional, but is not integrally closed. The ring O has at least one nonzero prime ideal that's not invertible and some nonzero ideal in O does not have a unique prime ideal factorization, since otherwise O would be a Dedekind domain and thus would be integrally closed. We will define a special ideal in O, called its conductor, that is closely related to the noninvertible prime ideals in O. The nonzero ideals in O that are relatively prime to the conductor are invertible in O and have unique factorization into prime ideals in O. Definition 1.1. The conductor of an order O in the number field K is c = cO = fx 2 K : xOK ⊂ Og: This is a subset of O since 1 2 OK , so c = fx 2 OK : xOK ⊂ Og = fx 2 O : xOK ⊂ Og; and the last formula for c shows us that c is the annihilator of OK =O as an O-module: c = AnnO(OK =O): Example 1.2. Let K = Q(i) and O = Z[2i] = Z + Z2i. For x = a + 2bi in O, where a and b are integers, we have xZ[i] ⊂ O if and only if xi 2 O, which is equivalent to a being even. Therefore c = f2m + 2ni : m; n 2 Zg. 1 Writing the condition x 2 c as OK ⊂ x O when x 6= 0, the conductor is the set of all common denominators when we write the algebraic integers of K as ratios from O, together with 0. Since K is the fraction field of O and OK is a finitely generated O-module (because it is a finitely generated Z-module) there are such common denominators, so c 6= f0g. Explicitly, since the index m = [OK : O] is finite, mOK ⊂ O, so m 2 c. The conductor is an ideal in O. Since O ⊂ OK , we have 1 2 c if and only if O = OK . So c is a proper ideal in O when O is a nonmaximal order. The conductor of O is also an ideal in OK : if x 2 c and α 2 OK then αxOK ⊂ xOK ⊂ O, so αx 2 c. A nonzero ideal of one ring that is also an ideal of a larger ring might seem a bit peculiar. (The principal ideals (3) in Z[i] and Z[2i] are written the same way but are not the same; the first contains 3i and the second does not.) The property of c being an ideal in both O and OK leads to a characterization of it: Theorem 1.3. An ideal in OK lies in O if and only if it is contained in c, so c is the largest ideal of OK that is contained in O. Proof. If a is an ideal in OK that lies in O then aOK = a ⊂ O, so a ⊂ c. Conversely, an ideal in OK that is contained in c is contained in O since c ⊂ O. 1 2 KEITH CONRAD Remark 1.4. While every ideal of OK that lies in O is inside c, not all ideals of O that lie in c are ideals of OK . When O = Z[2i] = Z + Z2i, the ideal 2O = Z2 + Z4i of O is contained in c = Z2 + Z2i but 2O is not an ideal in Z[i] since it contains 2 but not 2i. 2. Examples Example 2.1. Let K be a number field, a be a nonzero ideal in OK and set O = Z + a, so a ⊂ O ⊂ OK . Letting a \ Z = aZ, we will show c = dZ + a where d is a certain factor of a. Since a ⊂ O we have a ⊂ c by Theorem 1.3. By the ring isomorphism O=a = (Z + a)=a =∼ Z=aZ, each ideal in O containing a is dZ + a where d divides a. Therefore c = dZ + a for some d dividing a. What is this d? Since a \ Z = aZ, d is the smallest positive integer in c. The condition d 2 c is the same as dOK ⊂ O, and such integers d are the multiples of the exponent of OK =O (the exponent of a finite abelian group is the least integer annihilating the whole group). So as d we can use the exponent of OK =O: O = Z + a =) cO = dZ + a; where d = exponent of OK =O: + As a special case, consider a = cOK with c 2 Z and take K 6= Q, so n = [K : Q] ≥ 2. We will show the order O = Z + cOK has conductor cOK . To determine the exponent of Ln OK =O, write OK = i=1 Zei, where e1 = 1 (there is always some Z-basis of OK containing Ln ∼ Ln ∼ n−1 1.) Then O = Z ⊕ i=2 Zcei, so OK =O = i=2 Z=cZ = (Z=cZ) , which as an abelian group has exponent c (since n ≥ 2), so O = Z + cOK =) cO = cZ + cOK = cOK : ∼ This conductor is a principal ideal in OK but it is not principal as an ideal in O: O=cOK = n n Z=cZ while each nonzero α 2 cOK has norm a multiple of c , so [O : αO] ≥ c > c. The index formula [Z + cOK : cOK ] = c tells us when c = p is prime that pOK is a prime ideal in Z + pOK . When K is a quadratic field, every order in K has the form Z + cOK for a unique c ≥ 1. The conductor of this order is cOK and c = [OK : Z + cOK ] (because [K : Q] = 2). It is standard to label the conductor as c rather than cOK . This is special to the quadratic case. That is, for orders in quadratic fields the label \conductor" and \index" (in OK ) mean the same thing. For example, the order of conductor 2 in Q(i) is Z[2i] and the order p p p 1+ 5 of conductor 2 in Q( 5) is Z[2 2 ] = Z[ 5]. Orders of the form Z + aOK with a 2 Z − f0g have conductors that are principal ideals in OK . To find a conductor that is not a nonprincipal ideal in OK , we look at cubic orders. p p 3 3 3 Example 2.2.p Let K = Q( 7). The ring of integers is OK = Z[ 7] and (3) = p , where p = (3; 1 − 3 7). Set O = Z + p2. Since p2 \ Z = 3Z, O=p2 =∼ Z=3Z, so [O : p2] = 3, which means p2 is a prime ideal in O. (It looks a bit strange to say p2 is a prime ideal, but we're in the ring O, where p isn't even 2 2 a subset.) Since [OK : p ] = N(p) = 9, [OK : O] = 3, a prime, so OK =O has exponent 3. 2 2 Byp Example 2.1, the conductor of O is c = 3Z +p p = p . This is not a principal ideal in Z[ 3 7] since it has norm 9 and no element of Z[ 3 7] has norm ±9: p p 3 3 3 3 3 NK=Q(a + b 7 + c 49) = a + 7b + 49c − 3 · 7abc; p p so if a + b 3 7 + c 3 49 has norm ±9 we reduce mod 7 to get a3 ≡ ±9 mod 7, which has no solution. THE CONDUCTOR IDEAL 3 p Example 2.3. Let K = Q( 3 19). The ring of integers is p p 2 p 1 + 3 19 + 3 19 (2.1) O = Z + Z 3 19 + Z : K 3 p We will compute the conductor c for the order Z[ 3 19]. This order has not been constructed as Z + a, so we can't appeal to Example 2.1 to compute c. Instead we will compute c using the definition of the conductor. For x = p p p p 3 3 2 3 3 a + b 19p + c 19 in Z[ 19], to have x 2 c means xOK ⊂ Z[ 19], which is equivalent to 3 xei 2 Z[ 19], where e1; e2; e3 is a Z-basis of OK . Using the visible Z-basis of OK in (2.1) leads to the conditions a; b; c, and (a + b + c)=3 2 Z. Writing a + b + c = 3d, p 2 p p 2 p 2 x = a(1 − 3 19 ) + b( 3 19 − 3 19 ) + d · 3 3 19 ; so p 2 p p 2 p 2 (2.2) c = Z(1 − 3 19 ) + Z( 3 19 − 3 19 ) + Z · 3 3 19 : Expressing the Z-spanning set for c in (2.2) in terms of the Z-basis of OK , p 0 3 2 1 0 1 0 1 1 − 19 2 1 −3 p1 p p 2 3 B 3 19 − 3 19 C = @ 1 2 −3 A @ 19 A : @ p A p p 2 3 3 19 0 3 0 (1 + 3 19 + 3 19 )=3 The matrix has determinant 9, so [OK : c] = 9: the ideal c in OK has norm 9. We show by contradiction that c is not principalp in OpK . If c = αOK for some α 2 c, 3 3 9 = [OK : αOK ] = j NK=Q(α)j. Writing α = a + b 19 + c 19 with integers a; b, and c, 3 3 2 3 NK=Q(α) = a + 19b + 19 c − 3 · 19abc; which reduces mod 19 to a3. Since ±9 mod 19 is not a cube, we have a contradiction.
Recommended publications
  • Prime Ideals in Polynomial Rings in Several Indeterminates
    PROCEEDINGS OF THE AMIERICAN MIATHEMIIATICAL SOCIETY Volume 125, Number 1, January 1997, Pages 67 -74 S 0002-9939(97)03663-0 PRIME IDEALS IN POLYNOMIAL RINGS IN SEVERAL INDETERMINATES MIGUEL FERRERO (Communicated by Ken Goodearl) ABsTrRAcr. If P is a prime ideal of a polynomial ring K[xJ, where K is a field, then P is determined by an irreducible polynomial in K[x]. The purpose of this paper is to show that any prime ideal of a polynomial ring in n-indeterminates over a not necessarily commutative ring R is determined by its intersection with R plus n polynomials. INTRODUCTION Let K be a field and K[x] the polynomial ring over K in an indeterminate x. If P is a prime ideal of K[x], then there exists an irreducible polynomial f in K[x] such that P = K[x]f. This result is quite old and basic; however no corresponding result seems to be known for a polynomial ring in n indeterminates x1, ..., xn over K. Actually, it seems to be very difficult to find some system of generators for a prime ideal of K[xl,...,Xn]. Now, K [x1, ..., xn] is a Noetherian ring and by a converse of the principal ideal theorem for every prime ideal P of K[x1, ..., xn] there exist n polynomials fi, f..n such that P is minimal over (fi, ..., fn), the ideal generated by { fl, ..., f}n ([4], Theorem 153). Also, as a consequence of ([1], Theorem 1) it follows that any prime ideal of K[xl, X.., Xn] is determined by n polynomials.
    [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]
  • 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]
  • A Review of Commutative Ring Theory Mathematics Undergraduate Seminar: Toric Varieties
    A REVIEW OF COMMUTATIVE RING THEORY MATHEMATICS UNDERGRADUATE SEMINAR: TORIC VARIETIES ADRIANO FERNANDES Contents 1. Basic Definitions and Examples 1 2. Ideals and Quotient Rings 3 3. Properties and Types of Ideals 5 4. C-algebras 7 References 7 1. Basic Definitions and Examples In this first section, I define a ring and give some relevant examples of rings we have encountered before (and might have not thought of as abstract algebraic structures.) I will not cover many of the intermediate structures arising between rings and fields (e.g. integral domains, unique factorization domains, etc.) The interested reader is referred to Dummit and Foote. Definition 1.1 (Rings). The algebraic structure “ring” R is a set with two binary opera- tions + and , respectively named addition and multiplication, satisfying · (R, +) is an abelian group (i.e. a group with commutative addition), • is associative (i.e. a, b, c R, (a b) c = a (b c)) , • and the distributive8 law holds2 (i.e.· a,· b, c ·R, (·a + b) c = a c + b c, a (b + c)= • a b + a c.) 8 2 · · · · · · Moreover, the ring is commutative if multiplication is commutative. The ring has an identity, conventionally denoted 1, if there exists an element 1 R s.t. a R, 1 a = a 1=a. 2 8 2 · ·From now on, all rings considered will be commutative rings (after all, this is a review of commutative ring theory...) Since we will be talking substantially about the complex field C, let us recall the definition of such structure. Definition 1.2 (Fields).
    [Show full text]
  • Algebra 557: Weeks 3 and 4
    Algebra 557: Weeks 3 and 4 1 Expansion and Contraction of Ideals, Primary ideals. Suppose f : A B is a ring homomorphism and I A,J B are ideals. Then A can be thought→ of as a subring of B. We denote by⊂ Ie ( expansion⊂ of I) the ideal IB = f( I) B of B, and by J c ( contraction of J) the ideal J A = f − 1 ( J) A. The following are easy to verify: ∩ ⊂ I Iec, ⊂ J ce J , ⊂ Iece = Ie , J cec = J c. Since a subring of an integral domain is also an integral domain, and for any prime ideal p B, A/pc can be seen as a subring of B/p we have that ⊂ Theorem 1. The contraction of a prime ideal is a prime ideal. Remark 2. The expansion of a prime ideal need not be prime. For example, con- sider the extension Z Z[ √ 1 ] . Then, the expansion of the prime ideal (5) is − not prime in Z[ √ 1 ] , since 5 factors as (2 + i)(2 i) in Z[ √ 1 ] . − − − Definition 3. An ideal P A is called primary if its satisfies the property that for all x, y A, xy P , x P⊂, implies that yn P, for some n 0. ∈ ∈ ∈ ∈ ≥ Remark 4. An ideal P A is primary if and only if all zero divisors of the ring A/P are nilpotent. Since⊂ thsi propert is stable under passing to sub-rings we have as before that the contraction of a primary ideal remains primary. Moreover, it is immediate that Theorem 5.
    [Show full text]
  • 6. Localization
    52 Andreas Gathmann 6. Localization Localization is a very powerful technique in commutative algebra that often allows to reduce ques- tions on rings and modules to a union of smaller “local” problems. It can easily be motivated both from an algebraic and a geometric point of view, so let us start by explaining the idea behind it in these two settings. Remark 6.1 (Motivation for localization). (a) Algebraic motivation: Let R be a ring which is not a field, i. e. in which not all non-zero elements are units. The algebraic idea of localization is then to make more (or even all) non-zero elements invertible by introducing fractions, in the same way as one passes from the integers Z to the rational numbers Q. Let us have a more precise look at this particular example: in order to construct the rational numbers from the integers we start with R = Z, and let S = Znf0g be the subset of the elements of R that we would like to become invertible. On the set R×S we then consider the equivalence relation (a;s) ∼ (a0;s0) , as0 − a0s = 0 a and denote the equivalence class of a pair (a;s) by s . The set of these “fractions” is then obviously Q, and we can define addition and multiplication on it in the expected way by a a0 as0+a0s a a0 aa0 s + s0 := ss0 and s · s0 := ss0 . (b) Geometric motivation: Now let R = A(X) be the ring of polynomial functions on a variety X. In the same way as in (a) we can ask if it makes sense to consider fractions of such polynomials, i.
    [Show full text]
  • Prime Ideals in B-Algebras 1 Introduction
    International Journal of Algebra, Vol. 11, 2017, no. 7, 301 - 309 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.7838 Prime Ideals in B-Algebras Elsi Fitria1, Sri Gemawati, and Kartini Department of Mathematics University of Riau, Pekanbaru 28293, Indonesia 1Corresponding author Copyright c 2017 Elsi Fitria, Sri Gemawati, and Kartini. This article is distributed un- der the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we introduce the definition of ideal in B-algebra and some relate properties. Also, we introduce the definition of prime ideal in B-algebra and we obtain some of its properties. Keywords: B-algebras, B-subalgebras, ideal, prime ideal 1 Introduction In 1996, Y. Imai and K. Iseki introduced a new algebraic structure called BCK-algebra. In the same year, K. Iseki introduced the new idea be called BCI-algebra, which is generalization from BCK-algebra. In 2002, J. Neggers and H. S. Kim [9] constructed a new algebraic structure, they took some properties from BCI and BCK-algebra be called B-algebra. A non-empty set X with a binary operation ∗ and a constant 0 satisfying some axioms will construct an algebraic structure be called B-algebra. The concepts of B-algebra have been disscussed, e.g., a note on normal subalgebras in B-algebras by A. Walendziak in 2005, Direct Product of B-algebras by Lingcong and Endam in 2016, and Lagrange's Theorem for B-algebras by JS. Bantug in 2017.
    [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]
  • A Brief History of Ring Theory
    A Brief History of Ring Theory by Kristen Pollock Abstract Algebra II, Math 442 Loyola College, Spring 2005 A Brief History of Ring Theory Kristen Pollock 2 1. Introduction In order to fully define and examine an abstract ring, this essay will follow a procedure that is unlike a typical algebra textbook. That is, rather than initially offering just definitions, relevant examples will first be supplied so that the origins of a ring and its components can be better understood. Of course, this is the path that history has taken so what better way to proceed? First, it is important to understand that the abstract ring concept emerged from not one, but two theories: commutative ring theory and noncommutative ring the- ory. These two theories originated in different problems, were developed by different people and flourished in different directions. Still, these theories have much in com- mon and together form the foundation of today's ring theory. Specifically, modern commutative ring theory has its roots in problems of algebraic number theory and algebraic geometry. On the other hand, noncommutative ring theory originated from an attempt to expand the complex numbers to a variety of hypercomplex number systems. 2. Noncommutative Rings We will begin with noncommutative ring theory and its main originating ex- ample: the quaternions. According to Israel Kleiner's article \The Genesis of the Abstract Ring Concept," [2]. these numbers, created by Hamilton in 1843, are of the form a + bi + cj + dk (a; b; c; d 2 R) where addition is through its components 2 2 2 and multiplication is subject to the relations i =pj = k = ijk = −1.
    [Show full text]
  • Formal Power Series License: CC BY-NC-SA
    Formal Power Series License: CC BY-NC-SA Emma Franz April 28, 2015 1 Introduction The set S[[x]] of formal power series in x over a set S is the set of functions from the nonnegative integers to S. However, the way that we represent elements of S[[x]] will be as an infinite series, and operations in S[[x]] will be closely linked to the addition and multiplication of finite-degree polynomials. This paper will introduce a bit of the structure of sets of formal power series and then transfer over to a discussion of generating functions in combinatorics. The most familiar conceptualization of formal power series will come from taking coefficients of a power series from some sequence. Let fang = a0; a1; a2;::: be a sequence of numbers. Then 2 the formal power series associated with fang is the series A(s) = a0 + a1s + a2s + :::, where s is a formal variable. That is, we are not treating A as a function that can be evaluated for some s0. In general, A(s0) is not defined, but we will define A(0) to be a0. 2 Algebraic Structure Let R be a ring. We define R[[s]] to be the set of formal power series in s over R. Then R[[s]] is itself a ring, with the definitions of multiplication and addition following closely from how we define these operations for polynomials. 2 2 Let A(s) = a0 + a1s + a2s + ::: and B(s) = b0 + b1s + b1s + ::: be elements of R[[s]]. Then 2 the sum A(s) + B(s) is defined to be C(s) = c0 + c1s + c2s + :::, where ci = ai + bi for all i ≥ 0.
    [Show full text]
  • Formal Power Series Rings, Inverse Limits, and I-Adic Completions of Rings
    Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely many variables over a ring R, and show that it is Noetherian when R is. But we begin with a definition in much greater generality. Let S be a commutative semigroup (which will have identity 1S = 1) written multi- plicatively. The semigroup ring of S with coefficients in R may be thought of as the free R-module with basis S, with multiplication defined by the rule h k X X 0 0 X X 0 ( risi)( rjsj) = ( rirj)s: i=1 j=1 s2S 0 sisj =s We next want to construct a much larger ring in which infinite sums of multiples of elements of S are allowed. In order to insure that multiplication is well-defined, from now on we assume that S has the following additional property: (#) For all s 2 S, f(s1; s2) 2 S × S : s1s2 = sg is finite. Thus, each element of S has only finitely many factorizations as a product of two k1 kn elements. For example, we may take S to be the set of all monomials fx1 ··· xn : n (k1; : : : ; kn) 2 N g in n variables. For this chocie of S, the usual semigroup ring R[S] may be identified with the polynomial ring R[x1; : : : ; xn] in n indeterminates over R. We next construct a formal semigroup ring denoted R[[S]]: we may think of this ring formally as consisting of all functions from S to R, but we shall indicate elements of the P ring notationally as (possibly infinite) formal sums s2S rss, where the function corre- sponding to this formal sum maps s to rs for all s 2 S.
    [Show full text]
  • Algebraic Number Theory Course Notes (Fall 2006) Math 8803, Georgia Tech
    Algebraic Number Theory Course Notes (Fall 2006) Math 8803, Georgia Tech Matthew Baker E-mail address: [email protected] School of Mathematics, Georgia Institute of Technol- ogy, Atlanta, GA 30332-0140, USA Contents Preface v Chapter 1. Unique Factorization (and lack thereof) in number rings 1 1. UFD’s, Euclidean Domains, Gaussian Integers, and Fermat’s Last Theorem 1 2. Rings of integers, Dedekind domains, and unique factorization of ideals 7 3. The ideal class group 20 4. Exercises for Chapter 1 30 Chapter 2. Examples and Computational Methods 33 1. Computing the ring of integers in a number field 33 2. Kummer’s theorem on factoring ideals 38 3. The splitting of primes 41 4. Cyclotomic Fields 46 5. Exercises for Chapter 2 54 Chapter 3. Geometry of numbers and applications 57 1. Minkowski’s geometry of numbers 57 2. Dirichlet’s Unit Theorem 69 3. Exercises for Chapter 3 83 Chapter 4. Relative extensions 87 1. Localization 87 2. Galois theory and prime decomposition 96 3. Exercises for Chapter 4 113 Chapter 5. Introduction to completions 115 1. The field Qp 115 2. Absolute values 119 3. Hensel’s Lemma 126 4. Introductory p-adic analysis 128 5. Applications to Diophantine equations 132 Appendix A. Some background results from abstract algebra 139 iii iv CONTENTS 1. Euclidean Domains are UFD’s 139 2. The theorem of the primitive element and embeddings into algebraic closures 141 3. Free modules over a PID 143 Preface These are the lecture notes from a graduate-level Algebraic Number Theory course taught at the Georgia Institute of Technology in Fall 2006.
    [Show full text]