1993 IMO Problems/Problem 1
Let , where
is an integer. Prove that
cannot be expressed as the product of two non-constant polynomials with integer coefficients.
Solution
For the sake of contradiction, assume that for polynomials
and
in
. Furthermore, let
with
if
and
with
if
. This gives that
.
We have that , or
. WLOG, let
(and thus
). Since
and
divides
but not
, we need that
. We can keep on going up the chain until we get that
. Then, by equating coefficients once more, we get that
. Taking the equation
gives that
. This implies that
. Thus, the degree of
is at least
. However, because
is a non-constant factor of
, we have that the degree of
is at most
. Thus, the degree of
is
, implying that the degree of
is
.
From this fact, we have that there must exist a rational root of . The only candidates are
,
,
, and
, though.
,
,
, and
, so none of these work. Thus, there are no linear factors of
.
In other words, cannot be expressed as
for polynomials
and
in
. This means that
cannot be expressed as the product of two non-constant polynomials with integer coefficients.
Q.E.D.
Alternate Solution
Trivial by Perron's Criterion lol
Note: Quoting Perron's Criterion on the actual IMO will very likely result in a score in the set , since it was not a well-known result back then.