Number-Theoretic Menagerie

  • Colin Maclachlan
  • Alan W. Reid
Part of the Graduate Texts in Mathematics book series (GTM, volume 219)


This chapter gathers together number-theoretic concepts and results which will be used at various stages throughout the book. There are few proofs in this chapter and it should be regarded as a synopsis of some of the main results in algebraic number theory, the proofs and details of which can be found in one of the many excellent texts on algebraic number theory. Being labelled Chapter 0, the implication is that this is a reference section, and key results given in this chapter will be referred back to subsequently as required in the book. It is certainly not necessary for the reader to absorb all the material here before proceeding further. The basic ideas in Sections 0.1,0.2 and 0.3 will arise frequently in the succeeding chapters. However, until Chapter 6, only these basic ideas together with, in a couple of sections, some ideas from Sections 0.6,0.7 and 0.9 are required to understand the proofs and examples. Thus we suggest that the readers with a passing familiarity with basic notions in algebraic number theory should return to this chapter only when they encounter a concept with which they are unfamiliar.


Prime Ideal Number Field Valuation Ring Residue Field Algebraic Integer 


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Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Colin Maclachlan
    • 1
  • Alan W. Reid
    • 2
  1. 1.Department of Mathematical Sciences, Kings CollegeUniversity of AberdeenAberdeenUK
  2. 2.Department of MathematicsUniversity of Texas at AustinAustinUSA

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