In this article, I will prove both the Gauss primitive theorem and Eisenstein’s criterion using one proof method, and then try to generalize irreducibility criteria in . I think this article does a pretty neat job of unifying two criteria that most textbooks present as arbitrary and unrelated. As always, I claim this work to be completely original. I have not been influenced by any texts or other sources that I might have come across.
Gauss’s lemma states that
is primitive if the individual polynomials are primitive (they don’t have any non-unit constant factors dividing all the coefficients). Proof: The expanded polynomial is
.
Let us assume that divides all coefficients, where
is a prime element in
. There are
coefficients. If
, then it must divide at least one of
and
. Let us assume it divides
. Moving on to the coefficient of
, we see that for
to be true,
must divide at least one of
and
.
Continuing like this for coefficients, regardless of whether they are
or not, we see that at every step, we get a new
or
that is proved to be divisible by
. By the
th step, even in the worst case scenario, we have one polynomial such that all of its coefficients are divisible by
. This contradicts the assumption that both polynomials were primitive.
Now we move on to Eisenstein’s criterion: It states that for a polynomial
if ,
and
, then the polynomial is irreducible. Let us assume that the polynomial is reducible-
.
As , but
, we know that
divides only one of
and
. Let us assume that
. Now it is also true that
. This implies that
. Going on like this, we see that
divides every coefficient of
, which is a contradiction as it should not divide
. You may assume that
, and come upon an analogous contradiction.
Now how can we generalize Eisenstein’s theorem? For one, we may conclude that if and
, then the polynomial is irreducible too!
I will write down other generalizations when they come to mind.