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Local-Global Methods in Algebraic Number Theory

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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. 1 2 ZACHARY KIRSCHE 1.1. Absolute values and completions. Definition 1.1. Let K be a field. An absolute value or multiplicative valuation on K is a function j · j : K ! R such that the following properties hold: (1) For any x 2 K×, jxj > 0. Additionally, j0j = 0. (2) For all x; y 2 K, jxyj = jxj · jyj. (3) (Triangle Inequality) For all x; y 2 K, jx + yj ≤ jxj + jyj. If we further require that jx + yj ≤ maxfjxj; jyjg, we then say that j · j is a non-archimedean absolute value. This inequality is referred to as the ultrametric inequality. We will give a few examples of absolute values: Example 1.2. The standard Euclidean norm on Q gives an absolute value. We will denote this absolute value by j · j1, for reasons which will become clear soon. Example 1.3. Let p 2 Z be a prime. For x 2 Q we define νp(x) = ordp(x), n where we define ordp(x) for integers x to be the largest integer n such that p j x x and extend it to Q by ordp y = ordp(x) − ordp(y) for integers x and y. This is almost an absolute value, but it is a homomorphism from the additive group of Q to the multiplicative group of Z. To flip this into an absolute value, we define −νp(x) jxjp = p . This is called the p-adic absolute value on Q and is an example of a non-archimedean absolute value. Example 1.4. Let K=Q be an algebraic number field and let σ : K ! C be a complex embedding. We then define, for x 2 K, jxj = jσxjC, where j · jC is the standard Euclidean norm on C. Remark 1.5. The terminology \non-archimedean" derives from the fact that the Archimedean Property fails to hold with respect to the given absolute value. We recall that the Archimedean Property of R states that, for α; β 2 R, there is a positive integer n 2 N such that jnαj > jβj. However, if the absolute value j · j is non-archimedean, we may repeatedly apply the ultrametric inequality to see that jnαj ≤ jαj for any positive integer n, so the Archimedean Property fails to hold. The main importance of studying these absolute values lies in studying the com- pletions of various fields with respect to these absolute values. We briefly sketch the construction of the completion here: Construction 1.6. Let (K; j · j) be a field with an associated absolute value. We define the completion K^ as follows: the objects consist of equivalence classes of Cauchy sequences fang in K, where two such sequences fang and fbng are equiv- alent if limn!1 jan − bnj = 0. We can then define an absolute value on K^ by jfang − fbngj = limn!1 jan − bnj. It is important to note that K^ is a field and j · j is an absolute value on this field as defined above. This construction is most commonly used to construct the real numbers from Q by completing with respect to the Euclidean norm. However, as we have been looking at all the possible absolute values on Q, we are also interested in looking at all possible completions of Q. To this end, we define here another class of completions of Q: LOCAL-GLOBAL METHODS IN ALGEBRAIC NUMBER THEORY 3 Definition 1.7. Let p 2 Z be a prime. Define the field of p-adic numbers Qp to be the completion of Q with respect to j · jp. We will further develop some properties of the p-adic numbers in later sections. For now, we will focus on the problem of classifying all possible completions of an algebraic number field. 1.2. Classifying absolute values. We note that an absolute value on K trivially gives rise to a metric on K by d(x; y) = jx − yj. This allows us to study the field K as a topological space with the induced metric topology. We define an equivalence relation on absolute values by j · j1 ∼ j · j2 if and only if they induce the same topology on K. It is immediately apparent that two absolute values induce the same completion if and only if they are equivalent. With these definitions, one question that immediately comes up is one of clas- sifying absolute values on a field K up to equivalence. This problem is identical to the problem of identifying all possible completions of K. This problem was first studied in the case of the field Q, resulting in the following theorem due to Alexander Ostrowski: Theorem 1.8. (Ostrowski) Let j·j be an absolute value on Q. Then j·j is equivalent to one of the following: (1) The standard Euclidean absolute value j · j1. (2) The p-adic absolute value j · jp for some prime p. In particular, this implies that the only possible completions of Q are the real numbers R and the p-adic numbers Qp for each prime p. The proof of this theorem is somewhat technical and unenlightening, so for a full proof see [6]. We next consider the case of general algebraic number fields. We recall that, unlike in the case of the rational integers, we do not generally have unique factor- ization into prime elements in rings of integers OK for number fields, but rather we get unique factorization of ideals into prime ideals. Thus, a result similar to Ostrowski's theorem will hold, but with prime ideals instead of prime elements. Theorem 1.9. Let K=Q be an algebraic number field and OK its ring of integers. Then there is exactly one equivalence class of absolute values on K corresponding to each of the following: − ordp(x) (1) Each prime ideal p ⊆ OK , given by jxjp = jOK =pj . (We recall that ordp(x) is defined for elements x 2 OK as the largest integer n such that x 2 pn, and we define it for the entire number field as before by noting that K is the field of fractions of OK .) (2) Each real embedding σ : K ! R, given by jxj = jσxjR. (3) Each conjugate pair of complex embeddings σ : K ! C, given by jxj = jσxjC. We will discuss how to arrive at this result from Ostrowski's theorem near the end of the paper. Motivated by this result, we can give the following definitions: Definition 1.10. Let K=Q be a number field. An equivalence class of absolute values on K is called a place or a prime of K. If this equivalence class corresponds to a prime ideal, it is called a finite place. If instead it corresponds to a real or complex embedding of K, it is instead called an infinite place. The local fields of K are precisely the completions of K with respect to each place. 4 ZACHARY KIRSCHE Notation 1.11. Given a field K, we will denote its set of places by V . Given a place v 2 V , we will denote the completion of K with respect to v by Kv. In what sense these fields are \local" and why they are important will be the main focus of the remainder of this paper. Remark 1.12. It is possible to define local fields axiomatically; they are charac- terized primarily as fields which are complete and locally compact with respect to some absolute value. For the purpose of this paper, however, the above definition is more than sufficient. 1.3. Global fields. In this section, we will briefly discuss how we might generalize the theory of local fields to fields which are not algebraic number fields. As it turns out, the theory of local fields applies equally well to the case of function fields of certain algebraic curves. Though the remainder of this paper will focus exclusively on the case of algebraic number fields, it is nevertheless important to explore the more general theory of global fields.
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