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101 0 $aeng
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181   $ctxt$2rdacontent
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200 10$aNotes on Geometry and Arithmetic$b[electronic resource] /$fby Daniel Coray
205   $a1st ed. 2020.
210  1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2020.
215   $a1 online resource (XII, 181 p. 121 illus., 3 illus. in color.) 
225 1 $aUniversitext,$x0172-5939
311   $a3-030-43780-9 
327   $aChapter 1. Diophantus of Alexandria -- Chapter 2. Algebraic closure; affine space -- Chapter 3. Rational points; finite fields -- Chapter 4. Projective varieties; conics and quadrics -- Chapter 5. The Nullstellensatz -- Chapter 6. Euclidean rings -- Chapter 7. Cubic surfaces -- Chapter 8. p-adic completions -- Chapter 9. The Hasse principle -- Chapter 10. Diophantine dimension of fields.
330   $aThis English translation of Daniel Coray?s original French textbook Notes de géométrie et d?arithmétique introduces students to Diophantine geometry. It engages the reader with concrete and interesting problems using the language of classical geometry, setting aside all but the most essential ideas from algebraic geometry and commutative algebra. Readers are invited to discover rational points on varieties through an appealing ?hands on? approach that offers a pathway toward active research in arithmetic geometry. Along the way, the reader encounters the state of the art on solving certain classes of polynomial equations with beautiful geometric realizations, and travels a unique ascent towards variations on the Hasse Principle. Highlighting the importance of Diophantus of Alexandria as a precursor to the study of arithmetic over the rational numbers, this textbook introduces basic notions with an emphasis on Hilbert?s Nullstellensatz over an arbitrary field. A digression on Euclidian rings is followed by a thorough study of the arithmetic theory of cubic surfaces. Subsequent chapters are devoted to p-adic fields, the Hasse principle, and the subtle notion of Diophantine dimension of fields. All chapters contain exercises, with hints or complete solutions. Notes on Geometry and Arithmetic will appeal to a wide readership, ranging from graduate students through to researchers. Assuming only a basic background in abstract algebra and number theory, the text uses Diophantine questions to motivate readers seeking an accessible pathway into arithmetic geometry.
410  0$aUniversitext,$x0172-5939
606   $aAlgebraic geometry
606   $aAlgebraic Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M11019
615  0$aAlgebraic geometry.
615 14$aAlgebraic Geometry.
676   $a516.35
700   $aCoray$b Daniel$4aut$4http://id.loc.gov/vocabulary/relators/aut$01002778
801  0$bMiAaPQ
801  1$bMiAaPQ
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906   $aBOOK
912   $a996418257303316
996   $aNotes on Geometry and Arithmetic$92301702
997   $aUNISA