(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

determine whether the following polynomials are irreducible over Q,

i)f(x) = [itex]x^5+25x^4+15x^2+20 [/itex]

ii)f(x) = [itex]x^3+2x^2+3x+5 [/itex]

iii)f(x) = [itex]x^3+4x^2+3x+2 [/itex]

iv)f(x) = [itex]x^4+x^3+x^2+x+1 [/itex]

2. Relevant equations

3. The attempt at a solution

By eisensteins criterion let f(x) = [itex]a_n x^n+a_{n-1} x^{n-1}+...a_0[/itex]

if there exists p, a prime such that p does not divide [itex]a_n[/itex] , p divides [itex]a_{n-1}[/itex],...,p divides [itex]a_0[/itex] and [itex]p^2[/itex] does not divide [itex]a_0[/itex] then f(x) is irreducible over Q

So i) if p=5 => it is irreducible over Q

but not sure how to go about the others...

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# Factorization of polynomials

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