Theorem : the nth root of an integer
is either an integer or irrational.
Prove the more general result:
Theorem: If x is a root of the polynomial xm + c1 xm-1 + c2 xm-2+ ... + cm = 0,
(with the coefficients c1, c2, ... , cm all integers,
of course), then x is either an integer or irrational.
(Hint: suppose a/b is a root of the polynomial, and a and
b are relatively prime. Plug into the equation above and multiply
both sides by bm. Show that b must equal 1.)
Incidentally, this result naturally leads to the
definition of a quadratic integer .