Are These Polynomials Irreducible Over Q?

[gtfitzpatrick]

Homework Statement

determine whether the following polynomials are irreducible over Q,

i)f(x) = $x^5+25x^4+15x^2+20$
ii)f(x) = $x^3+2x^2+3x+5$
iii)f(x) = $x^3+4x^2+3x+2$
iv)f(x) = $x^4+x^3+x^2+x+1$

Homework Equations

The Attempt at a Solution

By eisensteins criterion let f(x) = $a_n x^n+a_{n-1} x^{n-1}+...a_0$
if there exists p, a prime such that p does not divide $a_n$ , p divides $a_{n-1}$,...,p divides $a_0$ and $p^2$ does not divide $a_0$ 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...

 

[gtfitzpatrick]

I think ii) and iii) are both reducible but iv) is irreducible as4-1 =3, a prime

 

[HallsofIvy]

Since ii and iii are cubics, if they were reducible, they would have to have at least one linear factor, so at least one rational root. By the "rational root theorem", any rational root to ii would have to be $\pm 1$ or $\pm 5$. Check whether any of those is a root. Similarly, any rational root to iii would have to be $\pm 1$ or $\pm 2$.

(Clearly neither ii nor iii has a positive root so you really only have two values to check in each problem.)

 

[micromass]

For (iv), the roots are easy to compute since

$$(x^5-1)=(x-1)(x^4+x^3+x^2+x+1)$$