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Introduction to the Real Spectrum

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Introduction to the Real Spectrum INTRODUCTION TO THE REAL SPECTRUM PETE L. CLARK 1. Introduction About eighteen months ago I was trying to learn a little bit about model theory for a student conference (the 2003 Arizona Winter School) explaining the connec- tions between model theory and arithmetic geometry. There are indeed deep and important connections here, and I was quickly convinced that model theory is an area of mathematics that it is good for everyone to know at least a little bit about, but that is not my point here. Rather, while reading a basic text, they cited as examples of basic algebraic struc- tures: group, ring, module, ¯eld, ordered ¯eld, . , and here I stopped: ordered ¯eld?!? Not since the tedious axiomatic description of R in my ¯rst undergraduate analysis class had I even encountered the term; what kind of loon would regard an ordered ¯eld as an object as basic as the others in the list? This talk (given in 12/04 at Colorado College) is a meditation on how wrong I was.1 After a brief review of the notion of an ordered ¯eld (it's what you think it is!), we will aim to engage { and to some extent, to answer { the following questions: ² Which ¯elds can be ordered? ² If a ¯eld can be ordered, in how many di®erent ways can it be ordered? ² What information about a ¯eld is encoded in the set of all of its orderings? ² Can we regard the set of orderings of a ¯eld as some kind of space? ² When the ¯eld is the ¯eld of functions of an algebraic variety over a real-closed ¯eld R, how is this space related to the set of R-points of the variety? 2. Ordered fields and formally real fields By an ordering < on a ¯eld F we mean a total ordering on the ¯eld which is compatible with the ¯eld axioms in the following sense: (1) x1 < x2; y1 < y2 =) x1 + y1 < x2 + y2: (2) x > 0; y > 0 =) xy > 0: Let us repackage these axioms slightly in a more convenient form. Namely, for any ordering, de¯ne the positive cone P := fx 2 F j x ¸ 0g2. Then P satis¯es the 1x3 was omitted in the talk itself. As of late December 2006, I have made some minor changes and additions. In the last two years I have not made any serious attempts to deepen my un- derstanding of this material or incorporate it into my own research, but I still ¯nd the subject striking and elegant, and I hope the reader will learn enough to feel the same way. 2Note that, unfortunately, zero is in the positive cone. The more pedantic terminology \non- negative cone" has not caught on. 1 2 PETE L. CLARK following properties (here we use that for a subset S ½ F , ¡S = f¡s j s 2 Sg): (a) P + P ½ P . (b) P ¢ P ½ P . (c) P [ (¡P ) = F . (d) P \ (¡P ) = 0. Remark: In the presence of (a) and (b), (d) is equivalent to P \ (¡P ) 6= F and also to ¡1 6 2P \ (¡P ). Conversely, it is immediate to check that starting with a subset P ½ F satisfy- ing the given properties, one gets an ordering by decreeing x < y i® y ¡ x 2 P . In the sequel, we will allow ourselves to do something that the analytic philosophers would not like, namely we will call P itself an ordering. Note that 1 2 P : indeed exactly one of 1 and ¡1 is in P , and if ¡1 were in P then (¡1) ¢ (¡1) = 1 would be in P , so it's de¯nitely 1 that's in P . Similarly, for any a 2 F , either a 2 P or ¡a 2 P , so either way (§a)2 = a2 2 P . From this it fol- 2 2 lows that any element of the form a1 +¢ ¢ ¢+an lies in P . Denoting the set of all sums of squares in F by §¤, we get that elements of §¤ must be positive in any ordering. Example 0: Of course the real numbers R with the usual < forms an ordered ¯eld. In fact, since for any nonzero real number a, exactly one of a and ¡a is a sum of squares, §¤ itself forms an ordering in R, so the usual ordering on R is the unique ordering. Example 1: If F0 ½ (F; P ) is a sub¯eld of an ordered ¯eld, then P0 := P \ F0 gives an ordering, i.e., a sub¯eld of an ordered ¯eld gets an induced ordering. Since R has plenty of sub¯elds, this gives lots of examples of ordered ¯elds. For in- stance, the rational ¯eld Q ½ (R; <) gets a canonical ordering. Using the platitude 1 1 P 1 a = a2 ¢ a, one checks that if a 2 ¤, so is a . Thus, since every positive integer is a sum of squares,3 also in Q one has the property that either x or ¡x is a sum of squares, so the usual ordering on Q is again the unique one. p Example 2: F = Q( 2). There are two embeddings F,! Rpwhichp di®er from each other by the nontrivialp automorphism of F , which carries 2 7! ¡ 2. (Alge- braically speaking, 2 just denotes one of the two roots of the polynomial X2 ¡ 2; except by choosing an embedding into R,p there is no preference given to one or the other root.) In one of these embeddings, 2 goes to the positive real number whose square is 2, and in the other one it goes to the negative real number whose square is 2. Thus the two embeddings give distinct orderings, and it is easy to check that these are the only two orderings of F . In general, the orderings on any number ¯eld correspond K correspond bijectively to the embeddings from K into R (\the real places"); in particular, they are ¯nite in number. p p 2 P Non-example 1: F = Q( ¡1). Since ¡1 = ( ¡1) 2 ¤, any ordering on F 3It is a famous theorem of Lagrange that every positive integer is a sum of at most four squares, but here it su±ces to note that the positive integer n can be represented as 12 +:::+12 (n times). INTRODUCTION TO THE REAL SPECTRUM 3 would have ¡1 in its positive cone, a contradiction. De¯nition: A ¯eld F is formally real if ¡1 is not a sum of squares of elements of F . Now, we may apply the same reasoning as in Non-example 1: Observation: A ¯eld F which admits an ordering is necessarily formally real. Non-example 2: No ¯eld of characteristic p > 0 is formally real, since ¡1 = 1+:::+1 ((p ¡ 1) times). So only ¯elds of characteristic zero can admit orderings. Non-example 3: No p-adic ¯eld is formally real. Indeed, it su±ces to look at the ¯elds Qp. If p ´ 1 (mod 4) then ¡1 is a square in Qp. If p is an odd prime, 2 2 then it is well-known that X + Y = ¡1 has a solution over Fp, and Hensel's Lemma lifts it to a solution in Zp, so ¡1 is a sum of two squares in Qp. A slightly more elaborate application of Hensel's Lemma shows that ¡1 is a sum of 4 squares over Q2. Remarkably, the converse of the above observation holds. Theorem 1. (Artin-Schreier) Any formally real ¯eld admits at least one ordering. We will sketch the proof, if only to emphasize that it does not in any sense construct an ordering on a formally real ¯eld. Rather, given a formally real ¯eld, what we can produce is a canonical preordering. A preordering is a subset T ½ F satisfying axioms (a), (b) and (d) for orderings, but in place of (c) the weaker: 0 P (c ) ¤ ½ T . P Now what is clear is that a ¯eld is formally real i® ¤ is a preordering on F . Moreover, this is the unique minimal preordering in any formally real ¯eld. The union of a chain of preorderings is itself a preordering, so Zorn's Lemma4 entitles us to a preordering which is not properly contained in any other preordering, and the point is to show that any maximal preordering does in fact satisfy the property T [ (¡T ) = F , so gives an ordering. For this we need: Lemma 2. For a preordering T ½ F and an element a 2 F , the following are equivalent: (i) the set T [a] = fx + ya j x; y 2 T g is a preordering; (ii) a is not in ¡T . Proof: Since no preordering can contain both a and ¡a, (i) =) (ii) is clear. Convsersely, suppose a is not in ¡T . We claim that ¡1 is not in T [a], which is enough, since if T [¡a] contained any nonzero element t and its additive inverse ¡t, 1 it would contain ¡t ¢ t ¢ t2 = ¡1. Now, if ¡1 = x + ya, then ¡ya = 1 + x is a nonzero element of T , so a = ¡y¡2(1 + x) 2 ¡T , a contradiction. Thus a maximal preordering T ½ F must satisfy T [¡T = F , i.e., T is an or- dering. This completes the proof of the Artin-Schreier theorem. In fact, we showed more: 4Added on 12/30/06: I presume that the Artin Schreier theorem requires the Axiom of Choice in the formal sense but have not been able to track down a reference. 4 PETE L. CLARK Corollary 3. Let F be a formally real ¯eld and x be any element of F which is not a sum of squares.
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