Question

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 : http://mathoverflow.net/questions/18094/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.

Answer 1

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?

Answer 2

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.