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A Friendly Introduction to Ideal Class Groups and an Interesting Problem

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A Friendly Introduction to Ideal Class Groups and an Interesting Problem A Friendly Introduction to Ideal Class Groups and An Interesting Problem Sutirtha Datta,F.Y.B.Sc(CMI) July 2019 1 Introduction In this article, we will mainly deal with one of the many interesting topics of Algebraic Number Theory : Class Numbers ( Red Alert:Please don't drag CMI Alg-2 topic on conjugacy classes in a group here) and we will see a classic prob- lem known as "Gauss's class number problem for imaginary quadratic fields”, which will eventually lead us to a sweet theorem named Heegner Theorem (credit goes to Harold Stark as well) I assume the reader to know basic notions of ring,ideals,field,field extensions etc ,else they can just have a glance on my Galois Theory article.Still I will give definitions of basic notions of UFD, PID and so on below. 2 Some Preliminary Concepts 2.1 Notion of Integral Closure Before I give the definitions,I would like to ask the reader to realize the similarity between the notion of an element being algebraic "over" an algebraic extension and that of integral "over" an integral extension while going through this section. Consider R is a subring of the commutative ring S with 1 = 1S 2 R. Then S is integral over R if every s 2 S is integral over R Now what does being integral over some ring mean? Definition 2.1 (Integral Element). : An element s 2 S is integral over R if s is root of a monic polynomial in R[x]. Definition 2.2 (Integral Closure). : The integral closure of R in S is the set of elements of S that are integral over R. 1 Now if it happens that the integral closure of R in S ,is equal to R itself we will call that R is integrally closed in S. At this point, we are ready to use concept of an integral ring extension to define the "integers" in extension field of the rational numbers Q. Suppose K/Q is a field extension. Definition 2.3. An element α 2 K is called an algebraic integer if α is integral over Z, in other words, α is root of some polynomial f 2 Z[x] Now the definition α being an algebraic integer seems difficult to check.So we state a theorem to make life easy Theorem 2.1. An element α in some field extension of Q is an algebraic integer () α is algebraic over Q and its minimal polynomial mα,Q(x) has integer coefficients. One side of this theorem follows from the definition.For the other implica- tion,just use Gauss' lemma. Now we introduce another term.Just like Definition 2.3 take K=Q , a field ex- tension. Definition 2.4. The integral closure of Z in K is called the ring of integers of K and is denoted by OK An algebraic integer is clearly algebraic over Q.Note that the the integers Z are algebraic integers in Q, and so we often refer them as "rational integers". 2.2 Principal Ideal Domain and UFD Definition 2.5 (Integral domain). A nonzero commutative ring is called inte- gral domain if there is no non-trivial zero divisors. Definition 2.6 (PID). PID is an integral domain in which every ideal is prin- cipal, i.e., can be generated by a single element. One common instance of PID is Z,the ring of integers. I suppose the reader is fundamental theorem of arithmetic. If so we are going to introduce the concept of UFD ( or UFO ? ),which is quite similar to that. In short, a UFD is an integral domain where every non-trivial non-unit element can be presented as a product of prime ( or irreducible) element, uniquely upto the orders and units.Keep in mind that prime and irreducible are slightly different ( why ? ) concepts,although in UFD both are same. Now we state the formal definition. Definition 2.7 (Unique Factorisation Doamin). A UFD is defined to be an integral domain R in which every non-zero element x of R can be written as a 2 product (an empty product if x is a unit) of irreducible elements pi of R and a unit u: x = up1p2pn; n ≥ 0 and this representation is unique in the following sense: If q1; :::; qm are ir- reducible elements of R and w is a unit such that: x = wq1q2:::qm; m ≥ 0 then m = n, and 9 a bijective map φ : f1; :::; ng ! f1; :::; mg such that pi is associated to qφ(i) for i 2 f1; :::; ng. This sequence can be useful : PID ⊂ UFD ⊂ GCD Domain ⊂ Integrally Closed Domain ⊂ Integral Domains. Thus every PID is a UFD. 3 Ideal Class Groups Before coming to class groups we first need to get used to a few more topics.We will first introduce field of fractions , which will be useful in the notion of fractional ideals 3.1 Field of Fractions Suppose R is a commutative ring.Recall that if a is neither zero nor a zero- divisor and ab = ac in R then b = c.Also we know that a zero-divisor a cannot be a unit in R.If a is a zeo-divisor we cannot always cancel a in the equation ab = ac.So it feels like R is subring of some larger ring Q in which every non- zero element of R that is not a zero-divisor is a unit. The rational numbers Q are constructed from the integers Z by adding inverses. n In fact a rational number is of the form d , where n and d are integers. Note that a rational number does not have a unique representative in this way,as you n pn can always produce: d = pd So really a rational number is an equivalence class of pairs [n; d], where two such pairs [n; d] and [m; b] are equivalent iff nb = md. In fact , if n; d 2 R where R is some integral domain, then the field of fractions n or quotient field: F rac(R) is defined to be the set of all fractions d . Formally we state: Definition 3.1. Let R be any integral domain. Let N be the subset of R × R such that the second coordinate is non-zero. Define an equivalence relation ∼ on N as follows : (a; b) ∼ (c; d) iff ad = bc: 3 Definition 3.2. The field of fractions of R is the set of equivalence classes, under the equivalence relation defined above. Given two elements [a; b] and [c; d] define [a; b] + [c; d] = [ad + bc; bd] and [a; b] × [c; d] = [ab; cd] With these rules of addition and multiplication F rac(R) becomes a field. Moreover there is a natural injective ring homomorphism φ : R! F rac(R); so that we may identify R as a subring of F rac(R).In a nutshell, the field of fractions of an integral domain is the smallest field in which it can be embedded. 3.2 Fractional Ideal Consider A is an integral domain and K be its field of fractions. By a fractional ideal a of A ,we mean a nonzero A-submodule a of K such that there is a common denominator for a ,in other words, 9d 2 A; d 6= 0such that da ⊆ A.A non-zero ideal of A ,which is simply a fractional idela conatained in A,is also called an integral ideal . A fractional ideal I is called invertible if there is another fractional ideal J such that IJ = R (where IJ = f a1b1 + a2b2 + ::: + anbn : ai2 I , bi2 J , n is positive integerg is called the product of the two fractional ideals). In this case, the fractional ideal J is uniquely determined and equal to the generalized ideal quotient,or soemtimes what we say colon ideals: (R : I) = fx 2 K : xI ⊆ Rg: Now we will see the significance of this concept . If in an integral domain, every non-zero fractional ideal is invertible,then we call this a Dedekind Do- main , one of the core ideas in Algebraic Number Theory.Just to make things clear , ring of integers OK of a number field ( it is nothing but a finite degree field extension over Q)K is a Dedekind Domain. Lemme give another view point to Dedekind Domain.Let's go back to the relations of ring of algebraic integers andp UFD.Can we say the ring of al- gebraic integers isp UFD? Notep that ,in Z[ −5] , 6 has distinct factorisation :2:3 = 6 = (1 + −5)(1 − −5). But on the other hand, while passing to thep ideals,in Z[5],wep observe,p 6 factorp into product of prime factors , 6 = (2; 1+ −5)(2; 1− −5)(3; 1+ −5)(3; 1− −5) and this factorisation is unique. One can show that if OK is the ring of integers for some number field K then, for any nonzero idea I ⊂ OK ; I can be uniquely factored into product of prime ideals 3.3 Class Group and Class Numbers In simpler language, ideal class group is a measure of the extent to which unique factorization fails in the ring of integers of K.We state the formal definition as 4 follows: Definition 3.3. If R is an integral domain, then the quotient of the group of invertible fractional ideals of R by the subgroup of non-zero principal ideals of R is called class group of R. The order of the class group of R is called the class number of R.
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