Algebraic Numbers

  • Mahima Ranjan Adhikari
  • Avishek Adhikari


Chapter 11 introduces algebraic number theory which developed through the attempts of mathematicians to prove Fermat’s Last Theorem. An algebraic number is a complex number which is algebraic over the field Q of rational numbers. An algebraic number field is a subfield of the field C of complex numbers, which is a finite field extension of the field Q and obtained from Q by adjoining a finite number of algebraic elements. The concepts of algebraic numbers, algebraic integers, Gaussian integers, algebraic number fields and quadratic fields are introduced in this chapter after a short discussion on general properties of field extension and finite fields. There are several proofs of Fundamental Theorem of Algebra. It is proved in this chapter by using homotopy (discussed in Chap.  2). Moreover, countability of algebraic numbers, existence of transcendental numbers, impossibility of duplication of a general cube and that of trisection of a general angle are shown in this chapter.


Finite Field Fundamental Theorem Field Extension Algebraic Number Algebraic Integer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Adhikari, M.R., Adhikari, A.: Groups, Rings and Modules with Applications, 2nd edn. Universities Press, Hyderabad (2003) Google Scholar
  2. Adhikari, M.R., Adhikari, A.: Text Book of Linear Algebra: An Introduction to Modern Algebra. Allied Publishers, New Delhi (2004) Google Scholar
  3. Alaca, S., Williams, K.S.: Introductory Algebraic Number Theory. Cambridge University Press, Cambridge (2004) Google Scholar
  4. Artin, M.: Algebra. Prentice-Hall, Englewood Cliffs (1991) Google Scholar
  5. Birkhoff, G., Mac Lane, S.: A Survey of Modern Algebra. Universities Press, Hyderabad (2003) Google Scholar
  6. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, London (2008) Google Scholar
  7. Hungerford, T.W.: Algebra. Springer, New York (1974) Google Scholar
  8. Rotman, J.J.: An Introduction to Algebraic Topology. Springer, New York (1988) Google Scholar

Copyright information

© Springer India 2014

Authors and Affiliations

  • Mahima Ranjan Adhikari
    • 1
  • Avishek Adhikari
    • 2
  1. 1.Institute for Mathematics, Bioinformatics, Information Technology and Computer Science (IMBIC)KolkataIndia
  2. 2.Department of Pure MathematicsUniversity of CalcuttaKolkataIndia

Personalised recommendations