The splitting extention above Z_ (p)

Asked Aug 17 '17 at 12:49

Active Aug 17 '17 at 12:55

Viewed 30 times

F (x) is a polynomial on Z_ (p), and n is an irreducible polynomial. When a is the root of f (x), is the splitting field of f (x) on Z_ (p)?

Using Frobunius automorphism we have found that a, a ^ p, ... a ^ (p ^ (n-1)) is also the root of f (x). Since Z_ (p) is a perfect field, the splitting field of f (x) is also a separable extension. I want to show that the splitting field of f (x) is Z_ (p) (a) because a, a ^ p, ..., a ^ (p ^ (n-1) My guess is wrong or can you help me? Thank you for your hlp!

edited Aug 17 '17 at 12:55

asked Aug 17 '17 at 12:49

LeeHanWoong

1

You need to show that the other zeros are in $\Bbb{Z}_p(a)$. Can you manage that? – Jyrki Lahtonen Aug 17 '17 at 13:12
1

This is close to being a duplicate. – Jyrki Lahtonen Aug 17 '17 at 13:14
@JyrkiLahtonen Lahtonen Thank you! Z_p (a) is a finite field finite field and can be separated and has different roots. But are there any reasons why we have different things? – LeeHanWoong Aug 17 '17 at 13:29

The splitting extention above Z_ (p)

0 Answers0