I am having a little troubles with this problem, I think there is something subtle that I am missing

a) Show that where =

b) Find a minimal polynomial in where

c) Show if is a primitive element in , the degree of the minimal polynomial of is n

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- Dec 2nd 2009, 10:03 AMHavenMinimal Polynomials
I am having a little troubles with this problem, I think there is something subtle that I am missing

a) Show that where =

b) Find a minimal polynomial in where

c) Show if is a primitive element in , the degree of the minimal polynomial of is n - Dec 2nd 2009, 10:54 AMBruno J.
For . Suppose . Then . Now we must have and . But is impossible (why?).

For , I can tell you that the minimal polynomial will be but I'll let you prove it. - Dec 2nd 2009, 12:17 PMHaven
Okay, for c) i have, since is a primitive element, the

So,

So is a root of the polynomial

But this is of degree p*n and not n - Dec 2nd 2009, 06:56 PMaliceinwonderland
is a root of the polynomial over GF(p), but f(x) is reducible over GF(p) (check the case with p=2, n=2 s.t p^2=4).

Since is the splitting field of f(x) over GF(p) and it is the simple extension over GF(p) such that , the degree of the minimal polynomial of should be n.