Eisenstein's Criterion question.

[Omukara]

Hello,

I have a couple questions concerning Eisenstein's Criterion;

1) by making a substitution of the form x |-> x + a, show x^43 + 43x + 85 is irreducible over Q.

2) completely factorize into monic irreducible factors over Q for x^36 + 36x^8 - 405.I've only come across other examples to these which could be solved by recognizing that the polynomial resembled something similar to the binomial expansion of the form (x + 1)^n, but in these cases I cannot see how that would work.

Help would be, as always, much appreciated:)

(edit: sorry, only the first question is concerned with E's criterion, but help for either would be fantastic:])

 

[micromass]

Hi Omukara!

For (1), what happens to the polynomial after you substitute (x+1) for x. Does that polynomial satisfy Eisensteins criterion?

 

[Omukara]

hmm, I did consider (x+1)^43 + 43(x+1) + 85 at one point, but my lack of intuition makes it hard for me to understand why we make the substitution, since, if I'm not mistaken; if we take the prime p=43, immediately we can see E's criterion is satisfied.

Ah, I think I just had a light bulb moment after reading your comment a few more times:) we make the substitution to explicitly show that for all the coefficients of x of order between 42 and 1, E's criterion is satisfied since all such coefficients in (x+1)^43 will be a multiple of 43. So it follows that if it is irreducible for the substituion x+1, it is irreducible for x

Thank you so much!^_^EDIT: Just realized I wrote down the question incorrectly for question 2), the leading x was supposed to be of order 16, so it's more do-able now. Apologies!

 

[micromass]

EDIT: Just realized I wrote down the question incorrectly for question 2), the leading x was supposed to be of order 16, so it's more do-able now. Apologies!

That doesn't only make it more doable, that makes it quite easy! First, do a substitution y=x⁸. You'll obtain a quadratic equation, and these kind of equations are easy to factorize!