# Examples of nice families of irreducible polynomials over Z

Hi, i search for irreducible polynomials over Z which have variable coefficients you can "choose". Since I found nearly nothing in books or the internet i hope you can help me. Here 3 examples: Let g a polynomial over Z with degree smaller then n/2 ,then: $g* (\prod_{i=1}^n (x-a_i)) -1$ is irreducible if the a_i are all distinct. Here you can choose n the coefficients of g and the a_n so its a nice example. Another one,I found, is from Furtwängler : $x^4 (\prod_{i=1}^{n-4} (x-b_i)) -(-1)^n *(2x+4)$ where the b_i are strictly increasing.Can you generalize this example ?I think it should work also for some integers other than 2 and 4. Here is another nice example : Polynomial with the primes as coefficients irreducible?

I try to find examples where its easy to control zeros modulo p of the some irreducible polynomials and its derivation for another problem.

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Perron's criterion states that an integer polynomial $x^n + a_{n-1} x^{n-1} + ... + a_0$ is irreducible if $|a_{n-1}| > |a_{n-2}| + |a_{n-3}| + ... + |a_0| + 1$ (if I've gotten the statement correct) and $a_0 \neq 0$, and there are lots of ways to write down parameterized families of coefficients with this property.

But I am not really sure what you want, since already Eisenstein's criterion lets you write down large parameterized families of irreducible polynomials. Can you be more specific?

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hi,thank you!I know Perron.Eistenstein is not very useful because then the derivation modulo p is a bit boring,there should be cases where its 0 and not 0 for different x.I look at polynomials and their zeros modulo p^k and i have lemmas involving the derivation to tell me how the "lifting" of the zeros to a zero mod p^(k+1) might look like and how many there are.Its hard to tell in a short way and i know no book about it,just a short text in german. –  trew Nov 9 '10 at 18:03
It might be useful for me to have a polynomial not in the explicit form where you can use perron,but like polynomials in the examples,also it is a problem to have a very big coefficient,if you need to "kill" for example a term like the sum of the a_k to get a easier form mod p^k(see my first example: its usefull to elimate the sum of the a_k ) –  trew Nov 9 '10 at 18:10

If $m$ and $n$ are odd then $x^5+mx^2+n$ is irreducible over the rationals (as one finds out by reducing it mod 2).

Michael Filaseta (and various co-authors) has a string of results on families of irreducible polynomials. E.g., Filaseta, Finch and Leidy, T N Shorey's influence in the theory of irreducible polynomials, has results on irreducibility of Laguerre polynomials. Filaseta, Kumchev and Pasechnik, On the irreducibility of a truncated binomial expansion, does what the title says. Filaseta, Luca, Stanica and Underwood, Two diophantine approaches to the irreducibility of certain trinomials, proves that $x^{2p}+bx^p+c$ is irreducible whenever $p$ is an odd prime and $\gcd(p,q^2-1)=1$ for all prime $q$ dividing $181b$. There's more where that came from. If you have access to MathSciNet you might try typing in Filaseta and irreducible.

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See also papers by Bremner –  rwst Oct 29 '13 at 10:17
@rwst, Bremner has written a lot of papers. Your comment might be more helpful if (as I did in my answer) you point to a few papers in particular. It might be even better as a new answer, as that will be more visible than a comment on my answer. –  Gerry Myerson Oct 29 '13 at 11:13