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.