## Abstract

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.

## Keywords

Finite Field Fundamental Theorem Field Extension Algebraic Number Algebraic Integer## References

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