The Generalization and Proof of “Square Root of 2 Is Not a Rational Number” on the Integral Domain

Abstract

In this paper, the traditional proof of “square root of 2 is not a rational number” has been reviewed, and then the theory has been generalized to “if n is not a square, square root of n is not a rational number”. And then some conceptions of ring, integral domain, ideal, quotient ring in Advanced algebra, have been introduced. Integers can be regarded as an integral domain, the rational numbers can be regard as a fractional domain. Evens and odds are principal ideals in integral domain. The operations on evens and odds are operations on quotient ring. After introducing “the minimalist form” in fraction ring. The paper proves the main conclusion: in a integral domain, multiplicative subset S produces a fraction ring S−1R, and n is not a square element in R, then to every element a∈R, a2≠n.