Some Polynomial Irreducibility Criteria
Point of Post: In this post we discuss some (very few, about three) of the classic irreducibility criteria for polynomials, including Eisenstein’s criterion in its general form.
So, up until this point we’ve discussed Euclidean Domains, PIDs, and UFDs and (in our last set of posts) showed that certain polynomial rings are also UFDs and Euclidean domains. A central role in all three of these classes of rings (although particularly in UFDs) is the notion of irreducibility. Consequently, if we hope to put our theory to good use it would be helpful to have a list of easily checkable criteria for when a polynomial over some given ring is irreducible. In this post we create such a list (very short, about three), mostly for reference at a later date.
There are some basic one’s we’d like to begin with. Namely, in our last post we saw that a polynomial (for some field ) is divisible by (with ) if and only if . Now, note that if is degree two or three, then being reducible is equivalent to having a root since any factorization of a degree one or two or three polynomial will involve a factor of degree one. From this we get the following theorem:
Theorem: Let be a field and with . Then, is reducible in if and only if has a root in .
This theorem implies that is irreducible in and perhaps slightly less obviously that is irreducible in (since as a function it’s identically ).
Often times we want to prove something is irreducible in but it is more tenable (whether we want to check roots or something) to check it in some . Luckily we have the following:
Theorem: Let with leading coefficient not divisible by , a prime. If is irreducible in then is irreducible in .
Proof: Suppose that was reducible in , then Gauss’s lemma would tell us that is reducible in , say where , and since there is a natural homomorphism one would have that is reducible in . But, since the leading coefficient of is nonzero in we know that are still less than even after coefficients have reduced. Thus, we have reduced in which is impossible.
This in conjunction with the previous theorem allows us to conclude that and are irreducible in since they have no roots in and respectively.
We now come to the classic irreducibility, Eisenstein’s irreducibility criterion. It easily shows things such as or are irreducible in . Roughly, what the theorem says is that if you have a polynomial whose leading coefficient is not divisible by a prime, every other coefficient is divisible by this (same) prime, and the constant term is not divisible by this (same) prime then the polynomial is irreducible in , and so by Gauss’s lemma, irreducible in . Indeed:
Theorem(Eisenstein’s Irreducibility Criterion): Let be such that there exists some prime such that , , and then is irreducible in .
Proof: By Gauss’s lemma it suffices to prove that is irreducible in . So, suppose that . Now, consider the reduction morphism . We have then that
but note that for some (since each coefficient other than the leading one is divisible by ). But, since is a field we know that is a UFD and so and for . But, note that this implies that and so which is a contradiction.
This is the Eisenstein’s criterion that is usually encountered in a beginning undergraduate course, but thankfully having done material above this level should make clear that the above easily and fruitfully generalizes. Indeed, what was really used (secretly) was just that was a field, or more pertinently, an integral domain to get a contradiction to the fact that . Clearly we should be able to generalize this to the more general case. Indeed:
Theorem: Let be an integral domain and . Suppose that there exists a prime ideal such that , , and . Then, is irreducible in .
Proof: By Gauss’s lemma it suffices to prove that is irreducible in . To see this suppose that . Consider the reduction map . We can clearly deduce (just as in the previous problem then that) for some non-zero . We claim though that this implies . Indeed, since the constant term is equal to and is equal to zero we have that either or , assume that but . Note then that since the linear term of the product of the polynomials is and this must be zero, we may conclude from this and the fact that that (recall that is an integral domain). Next, note that the quadratic coefficient of the product is and applying similar logic allows us to conclude that . Continuing in this way we arrive at the conclusion that which is impossible since and is nonzero. This is a contradiction. Thus, which implies that which is a contradiction since . Thus, we could not have nontrivially factored in .
To see an example where this theorem may be useful consider the following corollary:
Corollary: Let be an integral domain and let . Then, is irreducible in .
Proof: We know that is a prime ideal in and , , and so Eisenstein’s criterion applies.
 Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.
 Rotman, Joseph J. Advanced Modern Algebra. Providence, RI: American Mathematical Society, 2010. Print.
 Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. Basic Abstract Algebra. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.