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Geometry: Euclid and Beyond - Robin Hartshorne - Google Books
<!DOCTYPE html><html><head><title>Geometry: Euclid and Beyond - Robin Hartshorne - Google Books</title><link rel="stylesheet" href="/books/css/_a33f2a89320471e58c940b9287b9d4eb/kl_viewport_kennedy_full_bundle.css" type="text/css" /><link rel="stylesheet"href="https://fonts.googleapis.com/css2?family=Product+Sans:wght@400"><script src="/books/javascript/v2_a33f2a89320471e58c940b9287b9d4eb__en.js"></script><script>_OC_Hooks = ["_OC_Page", "_OC_SearchReload", "_OC_TocReload", "_OC_EmptyFunc", "_OC_SearchPage", "_OC_QuotePage" ];for (var _OC_i = 0; _OC_i < _OC_Hooks.length; _OC_i++) {eval("var " + _OC_Hooks[_OC_i] + ";");}function _OC_InitHooks () {for (var i = 0; i < _OC_Hooks.length; i++) {var func = arguments[i];eval( _OC_Hooks[i] + " = func;");}}</script><link rel="canonical" href="https://books.google.com/books/about/Geometry_Euclid_and_Beyond.html?id=EJCSL9S6la0C"/><meta property="og:url" content="https://books.google.com/books/about/Geometry_Euclid_and_Beyond.html?id=EJCSL9S6la0C"/><meta name="title" content="Geometry: Euclid and Beyond"/><meta name="description" content="In recent years, I have been teaching a junior-senior-level course on the classi cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa rately. The remainder of the book is an exploration of questions that arise natu rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And in the last chapter we provide what is missing from Euclid's treatment of the five Platonic solids in Book XIII of the Elements. For a one-semester course such as I teach, Chapters 1 and 2 form the core material, which takes six to eight weeks."/><meta property="og:title" content="Geometry: Euclid and Beyond"/><meta property="og:type" content="book"/><meta property="og:site_name" content="Google Books"/><meta property="og:image" content="https://books.google.com.sg/books/content?id=EJCSL9S6la0C&printsec=frontcover&img=1&zoom=1&edge=curl&imgtk=AFLRE71GLrfcwpizStofLs6lMiYHhji2oPmPi0op-20ZFBSwjAPxLMYD3bhXmNRTgEqpDo5ric7lti4YNCHOC6eu-aLwtdLDDQ5iTjAQGDcaoA1UR9G-rPv07jO0_w_lWmDO7XP9LoZf"/><link rel="image_src" href="https://books.google.com.sg/books/content?id=EJCSL9S6la0C&printsec=frontcover&img=1&zoom=1&edge=curl&imgtk=AFLRE71GLrfcwpizStofLs6lMiYHhji2oPmPi0op-20ZFBSwjAPxLMYD3bhXmNRTgEqpDo5ric7lti4YNCHOC6eu-aLwtdLDDQ5iTjAQGDcaoA1UR9G-rPv07jO0_w_lWmDO7XP9LoZf"/><script></script><style>#gbar,#guser{font-size:13px;padding-top:1px !important;}#gbar{height:22px}#guser{padding-bottom:7px !important;text-align:right}.gbh,.gbd{border-top:1px solid #c9d7f1;font-size:1px}.gbh{height:0;position:absolute;top:24px;width:100%}@media all{.gb1{height:22px;margin-right:.5em;vertical-align:top}#gbar{float:left}}a.gb1,a.gb4{text-decoration:underline !important}a.gb1,a.gb4{color:#00c !important}.gbi .gb4{color:#dd8e27 !important}.gbf .gb4{color:#900 !important} #gbar { padding:.3em .6em !important;}</style></head><body class=""><div id=gbar><nobr><a target=_blank class=gb1 href="https://www.google.com.sg/search?tab=pw">Search</a> <a target=_blank class=gb1 href="https://www.google.com.sg/imghp?hl=en&tab=pi">Images</a> <a target=_blank class=gb1 href="https://maps.google.com.sg/maps?hl=en&tab=pl">Maps</a> <a target=_blank class=gb1 href="https://play.google.com/?hl=en&tab=p8">Play</a> <a target=_blank class=gb1 href="https://www.youtube.com/?tab=p1">YouTube</a> <a target=_blank class=gb1 href="https://news.google.com/?tab=pn">News</a> <a target=_blank class=gb1 href="https://mail.google.com/mail/?tab=pm">Gmail</a> <a target=_blank class=gb1 href="https://drive.google.com/?tab=po">Drive</a> <a target=_blank class=gb1 style="text-decoration:none" href="https://www.google.com.sg/intl/en/about/products?tab=ph"><u>More</u> »</a></nobr></div><div id=guser width=100%><nobr><span id=gbn class=gbi></span><span id=gbf class=gbf></span><span id=gbe></span><a target=_top id=gb_70 href="https://www.google.com/accounts/Login?service=print&continue=https://books.google.com.sg/books%3Fid%3DEJCSL9S6la0C%26q%3Dordered%2Bfield%26source%3Dgbs_word_cloud_r%26hl%3Den&hl=en&ec=GAZACg" class=gb4>Sign in</a></nobr></div><div class=gbh style=left:0></div><div class=gbh style=right:0></div><div role="alert" style="position: absolute; left: 0; right: 0;"><a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=ordered+field&source=gbs_word_cloud_r&hl=en&output=html_text" title="Screen reader users: click this link for accessible mode. 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For a one-semester course such as I teach, Chapters 1 and 2 form the core material, which takes six to eight weeks.","my_library_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fop%3Dlibrary\u0026hl=en","is_magazine":false,"is_public_domain":false,"last_page":{"pid":"PA530","order":545,"title":"530","h":789}},{"enableUserFeedbackUI":true,"pseudocontinuous":true,"is_cobrand":false,"sign_in_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fid%3DEJCSL9S6la0C%26q%3Dordered%2Bfield%26source%3Dgbs_word_cloud_r%26hl%3Den\u0026hl=en","isEntityPageViewport":false,"showViewportOnboarding":false,"showViewportPlainTextOnboarding":false},{"page":[{"pid":"PA137","highlights":[{"X":242,"Y":146,"W":42,"H":11},{"X":287,"Y":146,"W":24,"H":11},{"X":144,"Y":182,"W":24,"H":10},{"X":304,"Y":182,"W":41,"H":10},{"X":350,"Y":182,"W":24,"H":10},{"X":147,"Y":218,"W":24,"H":12},{"X":391,"Y":218,"W":42,"H":12},{"X":439,"Y":218,"W":24,"H":12},{"X":248,"Y":269,"W":24,"H":11},{"X":326,"Y":284,"W":23,"H":11},{"X":396,"Y":284,"W":41,"H":11},{"X":443,"Y":284,"W":23,"H":11},{"X":60,"Y":312,"W":42,"H":11},{"X":106,"Y":312,"W":24,"H":11},{"X":104,"Y":397,"W":24,"H":12},{"X":349,"Y":411,"W":38,"H":12},{"X":391,"Y":411,"W":22,"H":12},{"X":110,"Y":425,"W":39,"H":13},{"X":151,"Y":425,"W":23,"H":13},{"X":229,"Y":452,"W":24,"H":12},{"X":174,"Y":495,"W":24,"H":12},{"X":179,"Y":653,"W":42,"H":10},{"X":228,"Y":653,"W":24,"H":10},{"X":233,"Y":681,"W":41,"H":12},{"X":59,"Y":697,"W":24,"H":9},{"X":289,"Y":45,"W":38,"H":8}],"flags":8,"order":150,"vq":"ordered field"}]},null,{"number_of_results":85,"search_results":[{"page_id":"PA3","page_number":"3","snippet_text":"... \u003cb\u003efield\u003c/b\u003e ( which could be the real numbers , for example ) , and show that the Car- tesian plane formed of \u003cb\u003eordered\u003c/b\u003e pairs of elements of the \u003cb\u003efield\u003c/b\u003e forms a geometry satisfying our axioms . But a deeper investigation shows that the notion of\u0026nbsp;..."},{"page_id":"PA30","page_number":"30","snippet_text":"... \u003cb\u003efield\u003c/b\u003e of rational numbers Q , where points are \u003cb\u003eordered\u003c/b\u003e pairs of rational numbers , and let AB be the unit interval on the x - axis . Then the vertex C of the equilateral triangle , which would have to be the point ( 3 ) , actually does\u0026nbsp;..."},{"page_id":"PA71","page_number":"71","snippet_text":"... \u003cb\u003efield\u003c/b\u003e ( see definition in §14 ) . Take the set F2 of \u003cb\u003eordered\u003c/b\u003e pairs of elements of the \u003cb\u003efield\u003c/b\u003e F to be the set of points . Define lines to be those subsets defined by linear equations , as in Example 6.1.1 . Verify that the axioms ( 11 )\u0026nbsp;..."},{"page_id":"PA95","page_number":"95","snippet_text":"... fields , and then we will show more generally that the Cartesian plane over any \u003cb\u003eordered field\u003c/b\u003e satisfying a certain algebraic condition gives a model of Hilbert\u0026#39;s axioms ( 17.3 ) . The other most important model of Hilbert\u0026#39;s axioms is\u0026nbsp;..."},{"page_id":"PA117","page_number":"117","snippet_text":"... field . The axioms of incidence are valid over any field ( Section 14 ) . For the notion of betweenness we need an \u003cb\u003eordered field\u003c/b\u003e ( Section 15 ) . For the axiom ( C1 ) on transferring a line segment to a given ray , we need a property\u0026nbsp;..."},{"page_id":"PA119","page_number":"119","snippet_text":"Robin Hartshorne. We accept as given the \u003cb\u003efield\u003c/b\u003e of real numbers R. We call a point an \u003cb\u003eordered\u003c/b\u003e pair ( a , b ) of real numbers , and the set of all such \u003cb\u003eordered\u003c/b\u003e pairs is the Cartesian plane . As usual , we call the set of points ( a , 0 )\u0026nbsp;..."},{"page_id":"PA128","page_number":"128","snippet_text":"... \u003cb\u003efield\u003c/b\u003e , and based on this \u003cb\u003efield\u003c/b\u003e we will obtain a geometry . Thus , using a \u003cb\u003efield\u003c/b\u003e , we obtain a model of the abstract geometry determined by Hilbert\u0026#39;s axioms ... \u003cb\u003eordered\u003c/b\u003e 128 3. Geometry over \u003cb\u003eFields\u003c/b\u003e Abstract \u003cb\u003eFields\u003c/b\u003e and Incidence."},{"page_id":"PA129","page_number":"129","snippet_text":"... \u003cb\u003efield\u003c/b\u003e F , which may not be the real numbers , we need to make our definitions precise . Definition The plane II ( or II , if we want to indi- cate the \u003cb\u003efield\u003c/b\u003e ) , called the Cartesian plane over the \u003cb\u003efield\u003c/b\u003e F , is the set F2 of \u003cb\u003eordered\u003c/b\u003e\u0026nbsp;..."},{"page_id":"PA135","page_number":"135","snippet_text":"... turns out that this is not possible over an arbitrary field . We will have to impose some additional structure on the. 15 \u003cb\u003eOrdered Fields\u003c/b\u003e and Betweenness. 15. \u003cb\u003eOrdered Fields\u003c/b\u003e and Betweenness 135 \u003cb\u003eOrdered Fields\u003c/b\u003e and Betweenness."},{"page_id":"PA136","page_number":"136","snippet_text":"... field F , analogous to the usual notion of positive real numbers . This leads to the concept of an \u003cb\u003eordered field\u003c/b\u003e . Definition An \u003cb\u003eordered field\u003c/b\u003e is a field F , together with a subset P , whose elements are called positive , satisfying\u0026nbsp;..."},{"page_id":"PA137","page_number":"137","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e , where we take for P the positive rational numbers , in the usual sense . 15.2.2 The field of real numbers IR is an \u003cb\u003eordered field\u003c/b\u003e with the usual notion of positive elements . 15.2.3 The field of complex numbers C cannot be an\u0026nbsp;..."},{"page_id":"PA138","page_number":"138","snippet_text":"Robin Hartshorne. Now suppose conversely that F , P is a given \u003cb\u003eordered field\u003c/b\u003e . We define be- tweenness for points on a line as follows : Let A = ( a1 , a2 ) , B = ( b1 , b2 ) , C = ( C1 , C2 ) ... \u003cb\u003eordered field\u003c/b\u003e 138 3. Geometry over Fields."},{"page_id":"PA139","page_number":"139","snippet_text":"Robin Hartshorne. Proposition 15.4 Let F , P be an \u003cb\u003eordered field\u003c/b\u003e . Then the Cartesian plane II will satisfy ( A ) or ... ordered 15. \u003cb\u003eOrdered Fields\u003c/b\u003e and Betweenness 139."},{"page_id":"PA140","page_number":"140","snippet_text":"Robin Hartshorne. sense ) in R. Thus the study of Archimedean \u003cb\u003eordered fields\u003c/b\u003e is equivalent to the study of subfields of IR . See Section 18 for some examples of non - Archimedean \u003cb\u003eordered fields\u003c/b\u003e . Exercises 15.1 If a \u0026gt; 0 in an \u003cb\u003eordered field\u003c/b\u003e\u0026nbsp;..."},{"page_id":"PA141","page_number":"141","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e F are congruent if dist2 ( A , B ) = dist2 ( C , D ) . Since congruence is defined using the function dist2 from line segments to the field , the axiom ( C2 ) , transitivity of congruence , will be obvious . Notice that\u0026nbsp;..."},{"page_id":"PA142","page_number":"142","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e F are congruent if they have the same tangent , considered as an element of the set FU { o } . Because congruence is defined by a function with values in FU { } , axiom ( C5 ) , transitivity of congruence , becomes obvious\u0026nbsp;..."},{"page_id":"PA144","page_number":"144","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e F. Then the following conditions are equivalent : ( i ) II satisfies the circle - circle intersection property ( E ) . ( ii ) II satisfies the line - circle intersection property ( LCI ) . ( iii ) the field F satisfies\u0026nbsp;..."},{"page_id":"PA145","page_number":"145","snippet_text":"... \u003cb\u003eordered\u003c/b\u003e Pythagorean \u003cb\u003efield\u003c/b\u003e . Proof To show that £ is a \u003cb\u003efield\u003c/b\u003e , let a , b € . Then each of a , b can be expressed in a finite number of steps using rational numbers and operations + , - , . ,, c → √1 + c2 . Hence the same is true of a\u0026nbsp;..."},{"page_id":"PA146","page_number":"146","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e . Proof Similar to the proof to ( 16.3 ) . Note that we may take square roots only of positive elements . Since K is also a subfield of IR , we get the ordering on K from IR as above . Remark 16.4.1 We call this the\u0026nbsp;..."},{"page_id":"PA147","page_number":"147","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e ( without assuming Pythagorean or Euclidean ) . Let A , B be points of the associated plane II . Show that the circle г with center A and passing through B has infinitely many points on it . Hint : First do the case of the\u0026nbsp;..."},{"page_id":"PA148","page_number":"148","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e . You only have to check the trichotomy : If a 0 , and -a \u0026amp; P , consider P \u0026#39; = P + aP and use maxi- mality to show P \u0026#39; = P , so a e P. ( f ) Use the fact that F is algebraic over Q to show that the ordering is Archimedean\u0026nbsp;..."},{"page_id":"PA150","page_number":"150","snippet_text":"... \u003cb\u003efield\u003c/b\u003e . Theorem 17.2 Let F be an \u003cb\u003eordered\u003c/b\u003e Pythagorean \u003cb\u003efield\u003c/b\u003e ,. 150 3. Geometry over \u003cb\u003eFields\u003c/b\u003e."},{"page_id":"PA151","page_number":"151","snippet_text":"Robin Hartshorne. Theorem 17.2 Let F be an \u003cb\u003eordered\u003c/b\u003e Pythagorean \u003cb\u003efield\u003c/b\u003e , and let II be the associated Cartesian plane . Then ( ERM ) holds in II . Proof We think of II as having coordinates ( x , y ) . We will consider certain\u0026nbsp;..."},{"page_id":"PA153","page_number":"153","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e , then the Cartesian plane II over F is a Hilbert plane satisfying the parallel axiom ( P ) . The plane II will be Euclidean if and only if F is Euclidean . Proof We have previously verified the incidence axioms ( 11 )\u0026nbsp;..."},{"page_id":"PA156","page_number":"156","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e . In the Cartesian plane II over F , let I be the unit circle , and let E = ( -1 , 0 ) . Let a line 1 through E meet the circle at a point A. ( a ) If the line I has slope t , show that the coordinates of the point A are\u0026nbsp;..."},{"page_id":"PA157","page_number":"157","snippet_text":"Robin Hartshorne. 17.7 Let F be a field ( not necessarily an \u003cb\u003eordered field\u003c/b\u003e ) that does not contain a square root of -1 . In analogy to the situation above , we define the circle group of F to be the set C ( F ) FU { } with the operation\u0026nbsp;..."},{"page_id":"PA158","page_number":"158","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e . The elements of the field need not be numbers or distances . Any abstract field will do . We will take advantage of this abstraction to construct some non- Archimedean geometries . These examples will serve two functions\u0026nbsp;..."},{"page_id":"PA159","page_number":"159","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e , it remains to show that if e F , 90 , then either e P or - € P , but not both . Indeed , if 0 , then it is the quotient of two nonzero polynomials ø = f ( t ) / g ( t ) . Each of these has a finite number of zeros . If\u0026nbsp;..."},{"page_id":"PA160","page_number":"160","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e by taking as the positive elements P \u0026#39; = P \u0026#39; , and P \u0026#39; satisfies ( i ) and ( ii ) because P does . Lemma 18.3 Let F be a subset of Q \u0026#39; that is a field , and let w e F , √1 + w2 ¢ F. Then F \u0026#39; = { α + B√1 + w2 | α , ẞ € F }\u0026nbsp;..."},{"page_id":"PA161","page_number":"161","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e . Note again that \u0026#39; is not a field . But if 9 € C \u0026#39; , 9 \u0026gt; 0 , then √9 € C \u0026#39; also . Now we take K to be the set of all elements of \u0026#39; that can be obtained from R ( t ) by a finite number of operations + , - ,,, and \u0026gt; 0\u0026nbsp;..."},{"page_id":"PA162","page_number":"162","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e R ( t ) , arrange the following elements in increasing order : 0 , 1,5 , t , 1 / t , t + 1 , 1 / ( t + 1 ) , t - 1,2t , t2 - t , t2 − 1 , t + 1 , ( t − 1 ) / ( t + 1 ) . 18.2 Show that the field \u0026#39; of Proposition 18.2\u0026nbsp;..."},{"page_id":"PA163","page_number":"163","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e . ( Be careful not to use continuity . ) 18.9 Let F be any \u003cb\u003eordered field\u003c/b\u003e . Let F ( ( t ) ) be the set of Laurent series 9 ∞ = Σait , an # 0 , izn where the aЄ F and ne Z can be positive , zero , or negative . Define\u0026nbsp;..."},{"page_id":"PA165","page_number":"165","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e F ( Section 19 ) . Using this field F we can recover the usual theory of similar triangles ( Section 20 ) . To complete the circle , we show that if you start with a Hilbert plane II sat- isfying ( P ) , and if F is the\u0026nbsp;..."},{"page_id":"PA167","page_number":"167","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e with an infinite element t , then Euclid\u0026#39;s test would fail to distinguish between √2 and √2 + 1 / t . Having developed the theory of proportion abstractly in Book V , Euclid pro- ceeds to apply his theory to geometry in\u0026nbsp;..."},{"page_id":"PA168","page_number":"168","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e whose positive elements are the congruence classes of line segments . Here is where the concepts of modern abstract algebra play an essential role , because instead of using some preexisting notion of number , such as the\u0026nbsp;..."},{"page_id":"PA173","page_number":"173","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e F whose set of positive elements P is the set of congruence equivalence classes of line segments with operations + , · defined above . Proof This is a consequence of the purely algebraic lemma that follows . Lemma 19.4 Let\u0026nbsp;..."},{"page_id":"PA174","page_number":"174","snippet_text":"... \u003cb\u003eordered\u003c/b\u003e pair by an equivalent \u003cb\u003eordered\u003c/b\u003e pair , the result is equivalent ( ! ) . ( The symbol ( ! ) means a trivial ... \u003cb\u003efield\u003c/b\u003e . We define a mapping : P → F by a e P goes to ( a + b , b ) for any be P. This mapping is 1 - to - 1 onto\u0026nbsp;..."},{"page_id":"PA177","page_number":"177","snippet_text":"... \u003cb\u003eorder\u003c/b\u003e . Proposition 20.2 ( VI.2 ) In any triangle ABC , let B\u0026#39;C \u0026#39; be drawn parallel to BC . Then the sides AB and AC ... \u003cb\u003efield\u003c/b\u003e F , the fourth proportional to three given quantities is uniquely determined . Hence AD AC \u0026#39; . Since the\u0026nbsp;..."},{"page_id":"PA186","page_number":"186","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e F ( for example the real numbers ) , and making a geometry whose points are ordered pairs of elements of the field F. This is the Cartesian approach ( cf. Section 13 ) . In this model we defined lines and congruence\u0026nbsp;..."},{"page_id":"PA187","page_number":"187","snippet_text":"... field , with properties different from the geometries over fields . If we drop the parallel axiom , this is indeed ... \u003cb\u003eordered field\u003c/b\u003e of segment arithmetic in II ( 19.3 ) . Then F is Pythagorean ( 20.7 ) , and II is isomorphic to\u0026nbsp;..."},{"page_id":"PA189","page_number":"189","snippet_text":"... \u003cb\u003eordered\u003c/b\u003e pairs of the \u003cb\u003efield\u003c/b\u003e F ; so is 1 - to - 1 and onto . We must verify that is compatible with the notions of line , betweenness , congruence of segments , and congruence of angles . And remember that in II these are undefined\u0026nbsp;..."},{"page_id":"PA191","page_number":"191","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e F. By ( 15.4 ) , the plane II satisfies ( D ) if and only if the field F sat- isfies ( D \u0026#39; ) . And then by ( 15.5 ) , F ≈ R. Exercises 21.1 Given two adjacent nonoverlapping angles x , ẞ. 21. Introduction of Coordinates 191."},{"page_id":"PA199","page_number":"199","snippet_text":"... \u003cb\u003efield\u003c/b\u003e of segment arith- metic ( 24.7.3 ) . I do not know any purely geometric proof of this fact . In the non ... \u003cb\u003eorder\u003c/b\u003e to express them both as unions of congruent triangles . For each i , j consider the inter- section TS in P\u0026nbsp;..."},{"page_id":"PA205","page_number":"205","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e ( cf. Section 15 ) we define a \u0026gt; b if a − b ɛ P. Then the relation \u0026gt; has all the usual properties ( 15.2 ) . Definition A measure of area function on a Hilbert plane is a function a , defined on the set P of all figures\u0026nbsp;..."},{"page_id":"PA238","page_number":"238","snippet_text":"... \u003cb\u003eordered\u003c/b\u003e pair ( a , b ) and sends it to ( a + 2b , b - 4a ) , and regard a , b as elements of Z / 3Z . Start- ing ... \u003cb\u003efield\u003c/b\u003e extension Q ( z ) of degree 2 over Q. i Z D -13 Now , if a is a rational multiple of л , then we can write a\u0026nbsp;..."},{"page_id":"PA245","page_number":"245","snippet_text":"... \u003cb\u003efield\u003c/b\u003e theory , we will now derive a necessary and sufficient condition for a real number to be constructible . It ... \u003cb\u003eorder\u003c/b\u003e 2 \u0026quot; for some n . Proof Suppose a is constructible . Then there is a tower of \u003cb\u003efields\u003c/b\u003e Q = Fo F1 SSF CR F1 as\u0026nbsp;..."},{"page_id":"PA246","page_number":"246","snippet_text":"... \u003cb\u003efields\u003c/b\u003e ( F ) , so the splitting \u003cb\u003efield\u003c/b\u003e E of f ( x ) will be a subfield of F. Hence also the degree of E / Q , which is equal to the \u003cb\u003eorder\u003c/b\u003e of the Galois group G of f ( x ) , is a power of 2 . Conversely , let a IR , and assume that the\u0026nbsp;..."},{"page_id":"PA257","page_number":"257","snippet_text":"Robin Hartshorne. group of the splitting \u003cb\u003efield\u003c/b\u003e of the extension Q ( a ) will be a quotient of this one , and its \u003cb\u003eorder\u003c/b\u003e will be a power of 2. By ( 28.7 ) therefore , a is constructible . The proof of ( 29.2 ) given above actually\u0026nbsp;..."},{"page_id":"PA258","page_number":"258","snippet_text":"... \u003cb\u003efield\u003c/b\u003e of nth roots of unity , it is a normal \u003cb\u003efield\u003c/b\u003e extension of Q with Galois group Z , which is an abelian group of \u003cb\u003eorder\u003c/b\u003e ø ( n ) , the Euler ø - function ( 32.7 ) . Since the Galois group is abelian , Q ( a ) is a normal extension of\u0026nbsp;..."},{"page_id":"PA275","page_number":"275","snippet_text":"... \u003cb\u003efield\u003c/b\u003e F if and only if the Galois group of the minimal poly- nomial of a over F has \u003cb\u003eorder\u003c/b\u003e 2a3b for some a , b ≥ 0 . Proof If a is so constructible , they by ( 31.6 ) the \u003cb\u003efield\u003c/b\u003e F ( x ) has degree 2\u0026#39;3o over F , for some r , s≥ 0. The\u0026nbsp;..."},{"page_id":"PA276","page_number":"276","snippet_text":"... \u003cb\u003eorder\u003c/b\u003e q2 . Let σe K generate K / L . Then σ = tot1 will generate K / L . Let M be the subgroup of H generated by ... \u003cb\u003efield\u003c/b\u003e E of T is the intersection of the splitting \u003cb\u003efield\u003c/b\u003e with the real numbers . Now a Є E , and by the fundamental\u0026nbsp;..."},{"page_id":"PA286","page_number":"286","snippet_text":"... \u003cb\u003efield\u003c/b\u003e extension with Galois group G. Then ( a ) The \u003cb\u003eorder\u003c/b\u003e of G is equal to the degree of the extension E / F . ( b ) The only elements of E fixed under all elements of G are the elements of F. ( c ) There is a 1 - to - 1 inclusion\u0026nbsp;..."},{"page_id":"PA287","page_number":"287","snippet_text":"... \u003cb\u003efields\u003c/b\u003e of degree 3 over Q , namely Q ( VŽ ) , Q ( w√2 ) , and Q ( 22 ) . These correspond to the three subgroups { e , ( 12 ) } , { e , ( 13 ) } , { e , ( 23 ) } of \u003cb\u003eorder\u003c/b\u003e 2 of S3 . The \u003cb\u003efield\u003c/b\u003e Q ( √3 ) , which is a normal extension of Q\u0026nbsp;..."},{"page_id":"PA288","page_number":"288","snippet_text":"... \u003cb\u003eorder\u003c/b\u003e is a multiple of 5. Hence ( from abstract group theory ) , G contains an ele- ment of \u003cb\u003eorder\u003c/b\u003e 5. In S5 , the ... \u003cb\u003efield\u003c/b\u003e E for f ( x ) over the prime \u003cb\u003efield\u003c/b\u003e IFp . Then there is a 1 - to - 1 cor- respondence between the roots of f\u0026nbsp;..."},{"page_id":"PA291","page_number":"291","snippet_text":"... \u003cb\u003efield\u003c/b\u003e of n is this same \u003cb\u003efield\u003c/b\u003e Q ( S ) . From ( 32.3 ) we find that for each 1 \u0026lt; r \u0026lt; n with ( r , n ) = 1 , there is ... \u003cb\u003eorder\u003c/b\u003e p − 1 . Example 32.7.1 - - - n = 3. © 3 = x2 + x + 1. Its roots are ∞ = ( −1 + √ − 3 ) and w2\u0026nbsp;..."},{"page_id":"PA326","page_number":"326","snippet_text":"... \u003cb\u003efield\u003c/b\u003e of segment arithmetic . For the non - Euclidean case , we will use an \u003cb\u003eordered\u003c/b\u003e abelian group ( Section 23 ) whose elements are constructed out of finite sums of angles . To be precise , we proceed as follows . Recall first that in\u0026nbsp;..."},{"page_id":"PA333","page_number":"333","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e F , we have three ordered abelian groups nat- urally associated with F : the additive group ( F , + ) , the multiplicative group ( F \u0026gt; 0 , \u0026#39; ) , and the unwound circle group G of the Cartesian plane over F. We can ask\u0026nbsp;..."},{"page_id":"PA342","page_number":"342","snippet_text":"... order . Finally , if you have four points on a line , and you take signed distances ( + or - depending on a ... \u003cb\u003eordered field\u003c/b\u003e F. 37.1 Stereographic projection . In three- dimensional space , imagine our plane II and a circle I\u0026nbsp;..."},{"page_id":"PA345","page_number":"345","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e , we can still define inversion in a circle I by the same method as at the beginning of this section . If y is a circle with pr ( 7 ) = y , show that y still meets I in two points , even though we do not have the axiom ( E )\u0026nbsp;..."},{"page_id":"PA361","page_number":"361","snippet_text":"Robin Hartshorne. In \u003cb\u003eorder\u003c/b\u003e to discuss ( E ) in the Poincaré model , we first need to identify what is a P - circle ... \u003cb\u003efield\u003c/b\u003e has characteristic 0 , this is equivalent to x = y , i.e. , OC is con- gruent to OC \u0026#39; in the usual sense\u0026nbsp;..."},{"page_id":"PA362","page_number":"362","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e F. Proof Since P - lines and P - circles are all either usual circles or lines through O , and since betweenness is the same in the P - model as in the ambient Euclidean space , ( E ) in the P - model follows directly from\u0026nbsp;..."},{"page_id":"PA366","page_number":"366","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e F , unless otherwise noted . Proofs should be based on the Euclidean geometry of the Cartesian plane over F. In particular , do not use any of the results of Section 34 or Section 35 that depend on Archimedes \u0026#39; axiom\u0026nbsp;..."},{"page_id":"PA372","page_number":"372","snippet_text":"... field that need not be Euclidean . Let F be a Pythagorean \u003cb\u003eordered field\u003c/b\u003e , let de F , and let I be the circle x2 + y2 = d , which may be a virtual circle if √d \u0026amp; F ( Exercise 37.17 ) . We define the Poincaré model in r as in the text\u0026nbsp;..."},{"page_id":"PA373","page_number":"373","snippet_text":"... field of all those real numbers that can be expressed using rational_numbers and a finite number of operations + , - ,,, a → √1 + a2 , and a → √a2- √2 , provided that a2 - √2 \u0026gt; 0 . ( a ) F is a Pythagorean \u003cb\u003eordered field\u003c/b\u003e , d = √2\u0026nbsp;..."},{"page_id":"PA391","page_number":"391","snippet_text":"... order of the operations . Definition In order to define multiplication of ends , we first fix a line ... \u003cb\u003eordered field\u003c/b\u003e . Proof We have already seen ( 41.3 ) that ( F , + ) is an abelian group with identity 0 . From the\u0026nbsp;..."},{"page_id":"PA392","page_number":"392","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e . To show that F is Euclidean , let a be a positive end , let the line ( α , −α ) meet ( 0 , ∞ ) at A , and let B be the midpoint of OA . Then the line perpendicular to ( 0 , ∞ ) at B will have an end ẞ with the\u0026nbsp;..."},{"page_id":"PA415","page_number":"415","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e F. It follows that two Hilbert planes with ( P ) will be isomorphic , as abstract geometries , if and only if their associated fields are isomorphic , as \u003cb\u003eordered fields\u003c/b\u003e . In this section we will do the same thing for non\u0026nbsp;..."},{"page_id":"PA416","page_number":"416","snippet_text":"... field in Section 39 , and have used the ambient Cartesian geometry to investigate some of its properties . Then in ... \u003cb\u003eordered fields\u003c/b\u003e . Proof Of course , the set of ends in the plane II is uniquely determined . But in order to\u0026nbsp;..."},{"page_id":"PA422","page_number":"422","snippet_text":"... ordering . Hence F1 and F are iso- morphic as \u003cb\u003eordered fields\u003c/b\u003e . Corollary 43.3 If II is a hyperbolic plane with associated field of ends F , then II is isomorphic to the Poincaré model over the field F. Proof Indeed , the field F is\u0026nbsp;..."},{"page_id":"PA424","page_number":"424","snippet_text":"... \u003cb\u003efield\u003c/b\u003e F , the group G is just the additive ... \u003cb\u003efield\u003c/b\u003e F , it is the group of positive elements of the \u003cb\u003efield\u003c/b\u003e under multiplication ( F \u0026gt; 0 , ) using the multiplicative distance function ( 39.10 ) . We say that a subgroup M of an \u003cb\u003eordered\u003c/b\u003e\u0026nbsp;..."},{"page_id":"PA427","page_number":"427","snippet_text":"... \u003cb\u003efield\u003c/b\u003e of segment arithmetic to be commutative . The construction was general- ized by Schwan , and finally reached ... \u003cb\u003eorder\u003c/b\u003e relation . So Bachmann defined the notion of a \u0026quot; metric plane \u0026quot; in which you retain the properties of\u0026nbsp;..."},{"page_id":"PA431","page_number":"431","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e F , corresponding to a nonzero convex subgroup M of the group G = ( F \u0026gt; 0 , · ) , show that the following conditions are equivalent : ( i ) The Lotschnitt axiom holds in ПI 。. ( ii ) All elements of M are infinitesimal\u0026nbsp;..."},{"page_id":"PA432","page_number":"432","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e , and let M be a nonzero convex sub- group of either ( F , + ) or ( F \u0026gt; o , ) or the circle group of F ( Exercise 17.6 ) . Show that M cannot be Archimedean . 43.10 Use Theorem 43.7 to prove the theorem of Greenberg ( 1988 )\u0026nbsp;..."},{"page_id":"PA488","page_number":"488","snippet_text":"... \u003cb\u003efield\u003c/b\u003e of 2 elements to be the girls , and the 35 lines of 3 points each to be the rows . Then with a little care you can find five lines that fill the space and an automorphism σ of \u003cb\u003eorder\u003c/b\u003e 7 that cycles those five lines through the set\u0026nbsp;..."},{"page_id":"PA507","page_number":"507","snippet_text":"... \u003cb\u003efield\u003c/b\u003e л . See pi An . See alternating group 2ASA ( double - side - angle - side ) , 53 2SAS ( double - side - angle ... \u003cb\u003eordered\u003c/b\u003e , 205 , 212 , 326 , 327 absolute length , 366 , 380 abstract \u003cb\u003efields\u003c/b\u003e , 128-135 abstract geometry , 415\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA507\u0026vq=ordered+field"},{"page_id":"PA508","page_number":"508","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e is subfield of R , 139 Archimedean solids , 436 , 460 , 461 Archimedes \u0026#39; axiom , 4 , 70 , 115 , 158 , 167 for a field , 139 for angles , 319 implies Aristotle\u0026#39;s axiom , 324 in Cartesian plane , 139 in dissections , 215\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA508\u0026vq=ordered+field"},{"page_id":"PA509","page_number":"509","snippet_text":"... \u003cb\u003eordered field\u003c/b\u003e , 137 used in exterior angle theorem , 101 bicapped pentagonal antiprism , 459 bicapped square antiprism , 456 bidiminished icosahedron , 464 Billingsley , Henry , 1 bilunabirotunda , 469 , Plate XIX Birkhoff , George\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA509\u0026vq=ordered+field"},{"page_id":"PA510","page_number":"510","snippet_text":"... \u003cb\u003efield\u003c/b\u003e , 156 , 157 , 401 circle - circle intersection , 108 , 112 equivalent to ( LCI ) , 110 , 145 , 423 , 431 in ... \u003cb\u003eordered\u003c/b\u003e , 137 conchoid , 260 , 263 , 264 conclusion , 14 condition ( * ) , 142 condition ( ** ) , 144 condition\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA510\u0026vq=ordered+field"},{"page_id":"PA514","page_number":"514","snippet_text":"... \u003cb\u003efield\u003c/b\u003e abstract , 118 additive group of , 333 , 401 associated to hyperbolic plane , 388 Cartesian plane over , 71 ... \u003cb\u003eordered\u003c/b\u003e , 2 , 117 , 135-140 Pythagorean , 142 , 145 skew , 132 , 133 , 140 fifth postulate . See parallel\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA514\u0026vq=ordered+field"},{"page_id":"PA515","page_number":"515","snippet_text":"... \u003cb\u003efield\u003c/b\u003e , 117-163 solid , 437 taxicab , 89 , 90 three - point , 67 , 68 without numbers , 166 Gerling , Christian ... \u003cb\u003eordered\u003c/b\u003e abelian , 205 , 212 solvable , 275 symmetric , 69 symmetry , 469-480 unwound circle , 327 , 3 333 Gödel\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA515\u0026vq=ordered+field"},{"page_id":"PA516","page_number":"516","snippet_text":"... field , 128-135 incidence geometry , 66 , 71 independence of axioms , 69 , 158 , 161 index of subgroup , 473 inequality in \u003cb\u003eordered field\u003c/b\u003e , 136 of angles , 94 of line segments , 85 infinite element , 159 infinite number of parallel lines\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA516\u0026vq=ordered+field"},{"page_id":"PA519","page_number":"519","snippet_text":"... order of group , 245 order , 73 for angles , 94 for line segments , 86 linear , 81 of four points on line , 79 ordered abelian group , 205 , 212 , 326 , 327 of segment addition , 423 \u003cb\u003eordered field\u003c/b\u003e , 2 , 117 , 165 , 135-140 Archimedean\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA519\u0026vq=ordered+field"},{"page_id":"PA521","page_number":"521","snippet_text":"... field , 142 , 145 , 151 non - Archimedean , 159 , 163 Pythagorean \u003cb\u003eordered field\u003c/b\u003e , 345 Poincaré model over , 372 Pythagorean theorem , 8 , 42 , 46 , 203. See also ( 1.47 ) in Index of Euclid\u0026#39;s Propositions by dissection , 213 , 217-219\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA521\u0026vq=ordered+field"},{"page_id":"PA522","page_number":"522","snippet_text":"... field , 175 rational Cartesian plane , 89 rational net of points , 426 rational numbers , 30 , 89 , 118 , 136 as \u003cb\u003eordered field\u003c/b\u003e , 137 Cartesian plane over , 143 dyadic , 333 , 396 ray , 77 , 79 , 141 congruence of , 88 limiting parallel\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA522\u0026vq=ordered+field"},{"page_id":"PA524","page_number":"524","snippet_text":"... field , 279 , 284 , 285 , 292 square antiprism , 456 square dipyramid , 459 square root of 2 in field , 132 is irrational , 117 square root in \u003cb\u003eordered field\u003c/b\u003e , 140 of line segment , 123 , 171 construction of , 125 in constructions , 242\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA524\u0026vq=ordered+field"}],"search_query_escaped":"ordered field"},{});</script></div></div></div><script>(function() {var href = window.location.href;if (href.indexOf('?') !== -1) {var parameters = href.split('?')[1].split('&');for (var i = 0; i < parameters.length; i++) {var param = parameters[i].split('=');if (param[0] == 'focus') {var elem = document.getElementById(param[1]);if (elem) {elem.focus();}}}}})();</script>