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($x-c=y$).
closed as too localized by Felipe Voloch, Mark Sapir, Martin Brandenburg, Bruce Westbury, Torsten Ekedahl Nov 8 '11 at 12:39
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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.
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.