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 .