DOI: 10.14456/mj-math.2018.5 วารสารคณิตศาสตร์ MJ-MATh 63(694) Jan–Apr, 2018 โดย สมาคมคณิตศาสตร์แห่งประเทศไทย ในพระบรมราชูปถัมภ์ http://MathThai.Org [email protected] สมบตั ิหลกั มูลของฟี ลดอ์ นั ดบั Fundamental Properties of Ordered Fields ภัทราวุธ จันทร์เสงี่ยม Pattrawut Chansangiam Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Ladkrabang, Bangkok, 10520, Thailand Email: [email protected] บทคดั ย่อ เราอภิปรายสมบัติเชิงพีชคณิต สมบัติเชิงอันดับและสมบัติเชิงทอพอโลยีที่สาคัญของ ฟีลด์อันดับ ในข้อเท็จจริงนั้น ค่าสัมบูรณ์ในฟีลด์อันดับใดๆ มีสมบัติคล้ายกับค่าสัมบูรณ์ของ จ านวนจริง เราให้บทพิสูจน์อย่างง่ายของการสมมูลกันระหว่าง สมบัติอาร์คิมีดิสกับความ หนาแน่น ของฟีลด์ย่อยตรรกยะ เรายังให้เงื่อนไขที่สมมูลกันสาหรับ ฟีลด์อันดับที่จะมีสมบัติ อาร์คิมีดิส ซึ่งเกี่ยวกับการลู่เข้าของลาดับและการทดสอบอนุกรมเรขาคณิต คำส ำคัญ: ฟิลด์อันดับ สมบัติของอาร์คิมีดิส ฟีลด์ย่อยตรรกยะ อนุกรมเรขาคณิต ABSTRACT We discuss fundamental algebraic-order-topological properties of ordered fields. In fact, the absolute value in any ordered field has properties similar to those of real numbers. We give a simple proof of the equivalence between the Archimedean property and the density of the rational subfield. We also provide equivalent conditions for an ordered field to be Archimedean, involving convergence of certain sequences and the geometric series test. Keywords: Ordered Field, Archimedean Property, Rational Subfield, Geometric Series วารสารคณิตศาสตร์ MJ-MATh 63(694) Jan-Apr, 2018 37 1. Introduction to those of real numbers. In particular, every In the course of undergraduate ordered field is a metric space. In any mathematical analysis (e.g. [1-3]), the first ordered field, there is a subset isomorphic to thing to learn is the real number system, i.e., the rational numbers, called the rational sub- the real numbers together with fundamental field (see Section 2). We provide a simple algebraic/order/topological properties. We proof for the fact that the Archi-medean begin with two algebraic operations, name- property is equivalent to the density of the ly, the addition and the multiplications. rational subfield (see Section 3). Moreover, From abstract-algebra point of view, the real this property is equivalent several properties numbers with these operations constitute a involving convergence of certain sequences field. Then we discuss how to compare or and the geometric series test (see Section 4). order two real numbers, and how nice is the order relation related to algebraic opera- 2. Ordered Fields tions. In fact, the real numbers is an example In this section, we review fundamental of an ordered field. The last axiom for the properties of arbitrary ordered fields. The real numbers is the completeness axiom or axiomatic properties of an ordered field are the least-upper-bound property, making the modeled from the real numbers. real numbers a complete ordered field. Consequently, the set of real numbers Definition 1. A order on a set E is a binary possesses the Archimedean property and the relation ≺ on E with the following rational numbers is dense in the real line. properties: From these properties, we can discuss con- (i) Trichotomy property: For each x, y ∈ vergence of sequences and series for real E, one and only one of the following numbers, and limits, continuity, differen- statements hold: x ≺ y, x = y, y ≺ x. tiation, integration, and other analytical con- (ii) Transitivity: For each x, y, z ∈ E, if x cepts for real-valued functions. Therefore, it ≺ y and y ≺ z then x ≺ z. is significant to discuss properties of In this case, we say that E is an ordered set arbitrary ordered field. See more related with respect to the order ≺. discussions in [4-6]. In this paper, we focus on abstract Definition 2. An ordered field is a field (F, properties of ordered fields. We provide the +, ∂ ) which is also an ordered set (F, ≺) such definition and examples of ordered fields, that and discuss fundamental algebraic-order- (i) For each x, y, z ∈ F, if y ≺ z then x topological properties. In fact, the absolute + y ≺ x + z. value in ordered field has properties similar 38 สมบัติหลักมูลของฟีลด์อันดับ (ii) For each x, y, z ∈ F, if 0 ≺ x and 1, 1 + 1, 1 + 1 + 1, … 0 ≺ y then 0 ≺ x ∂ y. are all different, so it contains copies of . Here, 0 denotes the additive identity in the Thus, an ordered field necessarily must field F. contain an infinite number of elements. It The property (i) means that the order follows that every finite field cannot be relation is compatible with addition. The ordered consistently with its algebraic property (ii) can be replaced by (ii)′. For structures. The universal mapping property each x, y, z ∈ F, if x ≺ y and 0 ≺ z then x ∂ z of the quotient field implies that the ring ≺ y ∂ z. The property (ii) or (ii)′ means that monomorphism ç F can be extended to the order relation is compatible with a monomorphism ç F. Call the image of multiplication. The collection of conditions this map the rational subfield of F, denoted (i) and (ii) is equivalent to the conditions (i) by F . We also denote and (ii)′. F =+++{1, 1 1, 1 1 1, . } and FF=ßß-{0} ( F ) Example 3. The following examples are where -=-≠FF{|xx }. We can define ordered fields: nx and xn for each natural number n and 1) The real numbers with respect to xF≠ by x + x + … + x (n times) and the usual addition, the usual multi- xx x (n times), respectively. We can also plication, and the usual order. define open and closed intervals in a similar 2) The field ()x of real rational manner to those in the real line. From the functions in the form p(x)/q(x) where trichotomy property, we define p(x) and qx()ò 0are polynomials ||aaa=- max{,} for each aF≠ . with real coefficients, here for each Then the following properties hold for any fg,()≠ x we define fg if and abc,,≠ F with c > 0. only if f (k) < g (k) for all sufficiently ñ positivity: ||0,a í and ||a = 0 if large real numbers k. and only if a = 0. ñ -ÇÇ||aaa ||. Any subfield of an ordered field is an ñ ||acÇ if and only if -Çcac Ç. ordered field inheriting the algebraic and ñ multiplicativity: ||||||ab= a b ; in order structures. For example, the field of particular ||||aann= for any n≠ . rational numbers and the field of real ñ triangle inequality: algebraic numbers are ordered subfield of ||||||ab+Ç a + b. the real numbers. In the context of ordered field, the As a ring, every ordered field F always binomial expansion theorem and Bernoulli’s has characteristic zero since the elements inequality also hold by mathematical วารสารคณิตศาสตร์ MJ-MATh 63(694) Jan-Apr, 2018 39 induction. a constructive proof of (D). There is a non- We can equip an ordered field F with a constructive proof using (AP1), the well- topological structure as follows. For each ordering principle and a contradiction, e.g., aF≠ and e > 0, the e - neighborhood of a [7]. is given by The next theorem provides a simple Bae ()=≠ { x F :| x -< a |e }. proof of the fact (D) by using (AP1) and the From which one can define open sets, closed following property: sets, continuity, convergence, and another (AP2): F is not bounded above. topological/metric notions. In particular, every ordered field is a metric space. Theorem 4. In an ordered field (F, >), we have (AP1), (AP2), and (D) are mutually 3.Archimedean Property and Density of equivalent. the Rationals Subfield in Ordered Fields Proof. The equivalence between (AP1) and A fundamental fact in mathematical (AP2) in an ordered field is easy to see. analysis, is the density of the rational Suppose (AP1) holds. The idea to prove (D) numbers in the real line. We shall discuss is “partitioning with sufficiently small this property in an ordered field (F, <): spaces”. In order to get a fish, we first (D): Given xy, ≠ F with xy< , we can identify a “suitable place” it lives, then we find an r ≠ F such that xry<<. use a fishnet with sufficiently small meshes. This fact is equivalent to the fact that the Indeed, by (AP2) there are ab, ≠ F such topological closure of F in F is the whole that axybÇ<Ç. We shall partition the space F. A usual proof of (D) in textbooks interval [a, b] into many intervals so that x (see, e.g. [1, 2]) is given by expanding with and y belong to different intervals and the sufficiently large spaces. Indeed, if x and y endpoints of each interval are in F . The are elements of F with xy< , we will find proof is done if the length of each interval is mn, ≠ F such that xmny<< by scaling less than y-x. Indeed, (AP1) allows the the interval [x, y] to [nx, ny] where n is large existence of an n≠ F for which enough so that the interval [nx, ny] contains 1.nyx<- Now, we use 1/n as the length an element m≠ F . This task is done by of each interval. using a version of Archimedean property, Conversely, suppose (D) holds. Let x > 0 namely: . The density of F guarantees the existence (AP1): Given x > 0 , there is an n≠ F such of an r ≠ F such that 0 <<rx. With that 1.nx< rmn= where mn, ≠ , we have F Some authors (e.g. [3]) use (AP1) and 1.nmnxÇ< Thus, (AP1) holds. the well-ordering principle to give 40 สมบัติหลักมูลของฟีลด์อันดับ 4. Archimedean Property and The hypothesis (i) guarantees the existence of a natural number n such that 1/(cNe )< . Convergence of Certain Sequences and Hence, for any natural number nNí , we Series in Ordered Fields have Let us discuss the concept of 111 Archimedean property from two abstract 0|<==rrnn ||| Ç <<e . (1+ c ) n 1+ nc nc mathematical structures, namely, linearly n ordered groups and normed fields.