From: rgep@can.pmms.cam.ac.uk (Richard Pinch) Newsgroups: sci.math Subject: Re: irreducible polynomials dense in rationals Date: 27 Oct 1995 12:53:52 GMT Summary: Proof by direct construction Keywords: Eisenstein's criterion In article Laura Helen writes: >Suppose you want to show that irreducible polynomials over the rationals >are dense in all polynomials over the rationals. >[...] >So, if you can show that a polynomial in Q_p [x] which is irreducible >over Q_p is also irreducible over Q, you would have shown that >irreducible polynomials are dense in rational polynomials. Surely this is obvious? If there are factors over Q, they are also factors over Q_p? Or did I miss something? Here's another proof of the main result. Given a rational polynomial \$f = x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0\$, choose \$p\$ a large prime and write \$\$ g = (1+1/p^{n+1})x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0 \$\$ Then \$\$ p g(px) = (p^{n+1}+1) x^n + p^n a_{n-1} x^{n-1} + \cdots + p^2 a_1 x + p a_0 \$\$ If \$p\$ does not divide \$a_0\$ then this is irreducible by Eisenstein. So for \$p\$ sufficiently large, \$g\$ is irreducible and as close as desired to \$f\$. Richard Pinch; Queens' College, Cambridge