I have the polynomial

over .

Let be the splitting field of it.

How can I show that is divisible by 5.

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- April 1st 2009, 02:21 AMGaloisGroupDegree of extension.
I have the polynomial

over .

Let be the splitting field of it.

How can I show that is divisible by 5. - April 2nd 2009, 06:54 PMThePerfectHacker
- April 3rd 2009, 07:35 AMclic-clac
Hi ThePerfectHacker,

how do you prove the irreducibility of that polynomial? - April 3rd 2009, 09:43 AMMoo
Hello,

I know nothing of Galois theory, so I'm not too sure you're looking for rational solutions, real solutions, or complex solutions... (my guess goes for rational zeroes ^^)

Anyway, this may help : The Rational Roots Test - April 3rd 2009, 10:31 AMclic-clac
Hi moo,

looking for rational solutions would have given the answer for a polynomial in of degree 2 or 3 (in such case, "no solution irreducibility"). With higher degrees, the polynomial can be a product of irreducible polynomials and be reducible without having ant root. But thanks :) - April 3rd 2009, 10:34 AMMoo
- April 3rd 2009, 10:37 AMThePerfectHacker
If you use the rational zeros theorem you will realize it has no zeros. Therefore, if this polynomial is reducible then it means where . Expand RHS and compare coefficients and show that is impossible. This is a really painful way of proving irreducibility! Here is an easier way. Replace by and then . Now apply Einsteinstein irreducibility test with . (Nod)

I do not want to be insulting, but how many times do I have to tell you that not having zeros does not imply irreducibility! I think that is my third time telling you this already. (Smile)

Quote:

Stupid me >< I always miss that !

- April 3rd 2009, 10:50 AMclic-clac
Hopefully there was another proof than the painful one...:) Nice!

- April 3rd 2009, 11:28 AMGaloisGroup
The easiest way to show that f is irreducible is by using Eisenstein's Criterion at f(x-2) with p=5.