Still Completely Baffled by Splitting Fields

[JonJon]

I was asked to "calculate" the splitting field of x^p - x - c over Z_p (the integers mod p- a prime) where c is a nonzero element of Z_p. My first instinct upon working with this was that I thought the field couldn't exist because x^p = x mod p by fermat's little theorem. So it reduces to the splitting field of -c? Is there something that I am missing?

 

[daon2]

I was asked to "calculate" the splitting field of x^p - x - c over Z_p (the integers mod p- a prime) where c is a nonzero element of Z_p. My first instinct upon working with this was that I thought the field couldn't exist because x^p = x mod p by fermat's little theorem. So it reduces to the splitting field of -c? Is there something that I am missing?

Fermat only tells you here that \(\displaystyle f(x)\) has no roots in \(\displaystyle \mathbb{Z}_p\). The splitting field is an extension of \(\displaystyle \mathbb{Z}_p\) which contains the roots of \(\displaystyle f\). Of course if \(\displaystyle c=0\) then what Fermat's Little Theorem gives you is that the polynomial's roots are every element of your base field.

I'm supposing you need to find the field up to isomorphism. So assume \(\displaystyle S=\{a_1, a_2, ..., a_p\}\) are the roots. The splitting field is then \(\displaystyle \mathbb{Z}_p\). But it is actually a simple extension, which requires "noticing" something.