In: Advanced Math
Show that, for each natural number n, (x − 1)(x − 2)(x − 3). . .(x − n) − 1 is irreducible over Q, the rationals.
We show the irreducibility of the polynomial
over by using
the method of contradiction.
Note that the polynomial
and hence
is irreducible
over
if and
only if it is irreducible over
Now, let
if possible, there exist monic integer coefficient polynomials
of
degrees
such that
Then,
.
Since the polynomials
are integer coefficient polynomials
and hence
Now, the polynomial
is a polynomial of degre
(since
degrees of
are
and the degree
of
is
with
) and
Hence, is a polynomial
of degree at most
but it has
roots, namely,
which is possible only when the polynomial
is identically
which
implies that
which is a contradiction (since
are monic integer coefficient polynomials).
Hence it follows that the polynomial
is irreducible over