Produce an irreducible polynomial that can't be proved irreducible by using Eisenstein - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-20T01:13:27Z http://mathoverflow.net/feeds/question/80378 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80378/produce-an-irreducible-polynomial-that-cant-be-proved-irreducible-by-using-eisen Produce an irreducible polynomial that can't be proved irreducible by using Eisenstein david 2011-11-08T11:28:45Z 2011-11-08T12:32:26Z <p>give An example of an irreducible polynomial that cannot prove it by using the Eisenstein criterion even with the use of all linear change variable($x-c=y$). </p> http://mathoverflow.net/questions/80378/produce-an-irreducible-polynomial-that-cant-be-proved-irreducible-by-using-eisen/80380#80380 Answer by Gerry Myerson for Produce an irreducible polynomial that can't be proved irreducible by using Eisenstein Gerry Myerson 2011-11-08T11:37:07Z 2011-11-08T11:37:07Z <p>$x^2+8$ is an example. </p> http://mathoverflow.net/questions/80378/produce-an-irreducible-polynomial-that-cant-be-proved-irreducible-by-using-eisen/80382#80382 Answer by Maurizio Monge for Produce an irreducible polynomial that can't be proved irreducible by using Eisenstein Maurizio Monge 2011-11-08T12:28:11Z 2011-11-08T12:28:11Z <p>It is possible to produce a polynomial that cannot (provably) be proved to be irreducible considering the valuations of the roots, or even any polynomial function in the roots (which can be much more general than a linear substitution), for every possible valuation over the base field.</p> <p>Let $L/K$ be an unramifed extension of numeber fields (for instance the Hilbert class field of a $K$ with non-trivial class group), generated by $\alpha$ say. Then the minimal polynomial for $\alpha$ over $K$ will do.</p>