|San José State University|
& Tornado Alley
A Marvelous Lemma Due to Gauss|
Concerning the Roots of Algebraic Equations
with Integral Coefficients
Consider the algebraic equation
in which the coefficients ai are integers.
Lemma: Such algebraic equations cannot have a root that is rational but not an integer.
Suppose there is a root x=p/q in which p and q have no common factor. If the equation is multiplied by qn-1 the equation can be put into the form
The left-hand side of the above equation can only be an integer, but the right-hand side is an integer only if q=1.
The cube root of 2 cannot be rational since it is a solution to the algebraic equation x3−2=0 and it cannot be an integer.
To show that an expression, such as √3−√2, is not rational it is only a matter of finding an algebraic equation with integer coefficients for which it is a root. For the case of x=√3−√2, it immdiately follows that x² is 3−2√6+2 and hence
This is an algebraic equation with integer coefficient for which √3−√2 is a root. There is no integer equal to √3−√2. Therefore √3−√2 cannot be rational.
The failure of some expression to be a real rational number should not be construed to mean that it is a real irrational number. For example, the solution to
cannot be a real rational number. It is of course a complex number, of an integral nature.
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