Eisenstein
by math_explorer, Apr 14, 2011, 10:00 AM
If 
is an integer-coefficient polynomial and there's a prime 
with 
, 
for 
, 
, then the polynomial cannot be factored into polynomials with coefficients from 
(fancy notation: ![$\mathbb{Q}[x]$](//latex.artofproblemsolving.com/3/d/9/3d963b0224d7c8b822ad2df9d699fd043f0013e8.png)
).
(In fact, an integer-coefficient polynomial which factors nontrivially in also factors nontrivially in ![$\mathbb{Z}[x]$](//latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
, where "nontrivially" does not include factoring out an integer constant larger than 1. A short proof sketch: if 
after factoring out any nontrivial constants from 
, multiply 
and 
by the smallest numbers possible to make their coefficients integral, show that the result must be an integral multiple of , and show that no prime can divide that multiplier, so the result is exactly .)
hold on, now to use this thing
is an integer-coefficient polynomial and there's a prime 
with 
, 
for 
, 
, then the polynomial cannot be factored into polynomials with coefficients from 
(fancy notation: ![$\mathbb{Q}[x]$](http://latex.artofproblemsolving.com/3/d/9/3d963b0224d7c8b822ad2df9d699fd043f0013e8.png)
).(In fact, an integer-coefficient polynomial which factors nontrivially in
also factors nontrivially in ![$\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
, where "nontrivially" does not include factoring out an integer constant larger than 1. A short proof sketch: if 
after factoring out any nontrivial constants from 
, multiply 
and 
by the smallest numbers possible to make their coefficients integral, show that the result must be an integral multiple of , and show that no prime can divide that multiplier, so the result is exactly .)hold on, now to use this thing
This post has been edited 1 time. Last edited by math_explorer, Aug 19, 2011, 9:22 AM
May 2020
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