@MISC{ALGNUM_1.MIZ,
  AUTHOR = {Watase, Yasushige},
  TITLE = {Algebraic Numbers}
  SECTION1 = {Preliminaries},
  SECTION2 = {Extended Evaluation Function},
  SECTION3 = {Integral Element and Algebraic Numbers},
  SECTION4 = {Properties of Polynomial Ring over PID},
  DAY = {15},
  MONTH = {December},
  YEAR = {2016},
  ADDRESS1 = {Suginami-ku Matsunoki 6, 3-21 Tokyo\\Japan}
  SUMMARY = {This article provides definitions and examples upon an integral element of
  unital commutative rings. An algebraic number is also treated as consequence of a concept of
  "integral". Definitions for an integral closure, an algebraic integer and
  a transcendental numbers are included as well \cite{ZariskiSamuel:1975},
  \cite{AtiyaMacDonald:1969}.
  As an application of an algebraic number, this article includes a formal proof of
  a ring extension of rational number field $\mathbb{Q}$ induced by substitution
  of an algebraic number to the polynomial ring of $\mathbb{Q}[x]$ turns to be a field.}}

@BOOK{ZariskiSamuel:1975,
 AUTHOR={Zariski, Oscar and Samuel, Pierre},
 TITLE={Commutative Algebra I, 2nd Edition},
 PUBLISHER={Springer},
 YEAR=1975}

@BOOK{AtiyaMacDonald:1969,
 AUTHOR={Atiyah, Miechale Francis and MacDonald, Ian Grant},
 TITLE={Introduction to Commutative Algebra},
 PUBLISHER={Addison-Wesley},
 YEAR=1969}
