Non-Archimedian Fields. Topological Properties of Z p, Q p (p-adics Numbers)

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1 BULETINUL Uiversităţii Petrol Gaze di Ploieşti Vol. LVIII No. 2/ Seria Matematică - Iformatică - Fizică No-Archimedia Fields. Toological Proerties of Z, Q (-adics Numbers) Mureşa Alexe Căli Uiversitatea Petrol-Gaze di Ploieşti, Bd. Bucureşti 39, Ploieşti, Catedra de Matematică acmuresa@ug-loiesti.ro Abstract The reset wor tries to offer a ew aroach to some o-archimedia orms ad to some uusual roerties determied by these. The, as a examle, -adic umbers Q ad some toological roerties will be illustrated. The reset article will reset as a examle some abstract otios with -adic umbers exteded to Q durig the traslatio from o- Archimedia orms to the Q orm. Key words: -adics, o-archimedia, toological roerties No-Archimedia Absolute Value ad No-Archimedia s Proerties Fields Defiitio 1. A fuctio : E R + o a field E satisfyig: (1) (2) (3) x = 0 x = 0 xy = x y x + y x + y ( the Triagle Iequality) is called a orm or a absolute value. Defiitio 2. Let E be a fields with a absolute value. Let a E, r R +. The oe ball of radius r cetered at a is B r (a) = {x E : x a < r}. A close ball of radius r cetered at a is B r (a) = {x E : x a r}. A cloe ball of radius r cetered at a is a oe ad i the same time a close, ball. Defiitio 3. A absolute value is called o-archimedia if it satisfies a stroger versio of the Triagle Iequality: x + y max x, y x, y { } E ad archimedia otherwise; ad a field E with a o-archimedia absolute value is called a o-archimedia field. Proositio 1. A absolute value o Q is o-archimedia if ad oly if x 1 x N.

2 44 Mureşa Alexe Căli Proof. First suose is o-archimedia. The for itroductio N, let be P() : 1. For = 1 we have 1 = 1 1; P(1) is true. Cosiderig ow 1 for =. The + 1 max {, 1} = 1 1 ad P () P( + 1). Trough iductio, 1, N. Now we suose 1, N ad we wat to show a + b max { a, b }, a, b Q. If b = 0 we have a + b = a + 0 = a = max{ a, 0 } max { a, b }. So if we assume that b 0 the 1 a + b a a a a + b max{ a, b} / max ;1 + 1 max ; 1 b b b b b ad the it is eough to show that x + 1 max { x, 1}, x Q. For that, let be: = 0 If x 1 the x 1 for = 1,2,,. If x > 1 the x x for = 1,2,,. C x = 0 = 0 x + 1 = C x x. Ad i both cases, x ( + 1) max{ 1, x } = 0 The x 1 x ( + 1) max{ 1, x } ad because + = 0. ad x + 1 ( + 1) max{ 1, x } + 1 whe we have for the last relatios: 1 what we wat to show. x + 1 max { x, 1}, Also aother result which is ow is that: a absolute value o Q is archimedia if, y N such x > y. x Q, ( ) Lemma. I a o-archimedia field E if x,y E, x < y, the y = x + y. Proof. Assume x < x + y max y { x, y } by the iitial roreties x + y y. (1) Also, y = (x + y) x we have y max { x + y, -x } = max { x + y, x } ad because x < y for the last relatios y x + y. (2) Now for (1) ad (2) we have y = x + y.

3 No-Archimedia Fields. Toological Proerties of Z, Q (-adics Numbers) 45 Proositio 2. I a o-archimedia field E every triagle is isosceles. Proof. Let x, y, z be the verticus of the triagle a x y, y z, x z. If x y = y z we are doe. If x y y z the we assume y-z < x-y. But by Lemma, because y-z < x-y, we have x y = (x y) + (y z) = x z x y = x z ad the triagle is isosceles. Proositio 3. I a o-archimedia field. E, every oit i a oe ball is a ceter ad b B r (a) B r (a) = B r (b). Proof. Let b B r (a), so b a < r ad x ay elemet of B r (a). x b = (x a) + (a b) max { x a, a b } < r. Hece, x B r (b) sice x b < r B r (a) B r (b). Same B r (b) B r (a) is obviously ow. Therefore, B r (a) = B r (b). Corollary 1. Let E be a o-archimedia field. The for two ay oe balls or oe is cotaied i the other, or is either disjoit. Proof. We assume < r ad the roblem is: x B (a) ad x B r (a) B (a) B r (b) or B (b) B r (a). By Proositio 3, we have: x B r (a) B r (a) = B r (x) ad x B r (b) B r (b) = B r (x). The x B (a) = B (x) B r (x) = B r (b) sice < r. So, B (a) B r (b). Corollary 2. I a o-archimedia field every oe ball is cloe, that is a ball which is oe ad closed. Proof. Let B (a) be a oe ball i a o-archimedia field. Tae ay x i the boudary of B (a) B r (x) B (a) 0 for ay s > 0, so i articular, for s < r. By Corollary 1, sice B r (x) ad B (a) are ot disjoit, oe is coteied i the other ad because s < r we have x B r (x) B (a) B (a) is closed. So every oe ball is closed ad every ball is cloe. The -adic Absolute Value ad Toological Proerties for Z ad Q No-Archimedias Fields Defiitios. Let N be a rime, the for each Z, 0 we have = divides. for = 0 We defie f () = ad = α whe 0 Z, called the -adic absolute value. f () α with ot is a o-archimedia absolute value o

4 46 Mureşa Alexe Căli a a For Q, f = f (a) f (b) we defie a o-archimedia -adic absolute value i Q b b with the relatio: f ( x) x =, ( ) x Q ad f ( x y) x y = ( ) x, y Q. The we have: i N = α / α = ai, ai, N;0 ai 1 i= 0 are the -adic umbers from N, Z = = i α / α ai, ai N;0 ai 1 i= 0 are the -adic umbers from Z ad we ca defie these umbers as the iverses of aturals umbers writte i -base system ad the aturals umbers. Q = = i α / α ai, Z, ai N;0 ai i 1 i= are the -adic umbers from Q. Proositio 4. If E it s the comletio of a field E with resect to a absolute value. the E it s a field where we ca to exted the absolute value. Proof. See [3] of refereces. Proositio 5. Q is the comletio of Q with resect to Proof. See [3] of refereces. Proositio 6. Z,Q are comletes. Proof. See [3] of refereces.. Z = {x Q / x 1}. Proof. See [5] of refereces. Proositio 7. Z is comact ad Q is locally comact, by which we mea every x Q have a eighborhood which is comact. Proof. (1) We ow that Z is comlete, we wat to show that Z is totally bouded. Tae ay ε > 0 ad N such that - ε.. The a + Z = B - (a) B ε (a) because x a - < ε ( ) x a + Z. For a Z / Z the classes rerezetats is icluded i {0,1,,,, 1}. We have with defiitio of Z

5 No-Archimedia Fields. Toological Proerties of Z, Q (-adics Numbers) 47 Z U U U a + Z = B ( a) B ε a Z / Z a Z / Z a Z / Z ( a) a fiite cover of Z. So {B ε (a) : a Z / Z} is a fiite set of oe balls which cover Z, so Z is totally bouded, also is comlete ad result it is comact. (2) Let be f : Z Q ; f(y) = x + y for x Q ow, is cotiuous. Z is comact so the image of Z is x + Z is also comact. Now we ca observe that x + Z is a comact eighborhood of x Q because for z x +Z ; z x Z ; z x < 1, (Proositio 2). Remars We give, for studets, very commo defiitios, available i saces with a orm. Defiitios Let ( x ) be a sequece with etries i a field E with a absolute value.. 1. ( x ) is a Cauchy sequece if such that ( ) m N, x m x < ε., ( ) ε > 0, ( ) N N 2. If every Cauchy sequece i E coverges, E is comlete with resect to.. 3. We ca itroduce the equivalece relatio: x ) y ) if ( ) ε > 0, ( ) N N N ( (, x y ε such that ( ). ε < 4. The comletio of field E with resect to a absolute value. is the set : {[ ( x ) ]: ( x ) is a Cauchy sequeces o E} of all equivalece classes of Cauchy sequece with the equivalece relatio from 4) A oe cover of a set S is a family of oe sets { Si } such that i Si S A set S is called comact if every oe cover of S has a fiite subcover. If {S i}is ay oe cover of = S i S for some Ν. i 1 6. A set is called totally bouded if for ever ε > 0, there exist a fiite collectio of balls of radius ε which cover the set. Proositios 1. Comactess is reserved by cotiuous fuctios, what it meas, if f is cotiuous ad A is comact the f(a) is comact. 2. Ay closed subset of a comlete sace is comlete. 3. A set is comact if it is comlete ad totally bouded. ε ε Refereces 1. Colligwood, J., Mario, S., Mazowita, M. - P-adic umber, htt://adic.mathstat.uottawa.ca/~mat3166/reorts/-adic.df 2. Colojoara, I. - Aaliză matematică, Ed. Didactică şi Pedagogică, Bucureşti, 1983

6 48 Mureşa Alexe Căli 3. Gouvea, F.Q. - P-adic Numbers: A itroductio, Secod Editio, Sriger-Verlag, New Yor, K a t o, S. - Real ad -adic aalysis, Course otes for math 497c Mass rogram, fall 2000, Revised November 2001, Deartmet of mathematics, the Pesylvaia State Uiversity 5. Koblitz, N. - P-adic umbers, P-adic aalysis ad Z-eta Fuctios, Secod editio, Sriger- Verlag, New Yor, 1984 Rezumat Câmuri earhimediee. Prorietăţi toologice ale lui Z, Q (umere -adice) Lucrarea doreşte să dea o ouă abordare asura uor orme earhimediee şi asura uor ciudate rorietăţi iduse de acestea, aoi, ca exemlu, sut rezetate umerele -adice Q şi câteva rorietăţi toologice. Trecâd de la orme earhimediee la orma î Q se va da o exemlificare a uor oţiui abstracte î umerele di baza extise la Q.