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Suppose w is a primitive n-th root of unity, let r = Ord_n(p) (the smallest integer s.t. $p^r=1 mod n $), $f(w)=w^p$(the frobenius map), how to prove that $w,f(w),f^2(w),...f^{r-1}(w)$ are all different in Fp(w) so that the minimal monic polynomial of w over Fp factors into those r factors in Fp(w)?

Here if w is replaced with a primitive element of Fp(w) then it's easy to prove. But w is not a primitive element of Fp(w), it's a primitive element of Fp(w)/Fp.

Suppose w is a primitive n-th root of unity, let r = Ord_n(p) (the smallest integer s.t. $p^r=1 \pmod n $), $f(w)=w^p$(the frobenius map), how to prove that $w,f(w),f^2(w),...f^{r-1}(w)$ are all different in Fp(w) so that the minimal monic polynomial of w over Fp factors into those r factors in Fp(w)?

Here if w is replaced with a primitive element of Fp(w) then it's easy to prove. But w is not a primitive element of Fp(w), it's a primitive element of Fp(w)/Fp.

Suppose w is a primitive n-th root of unity, let r = Ord_n(p), $f(w)=w^p$(the frobenius map), how to prove that $w,f(w),f^2(w),...f^{r-1}(w)$ are all different in Fp(w) so that the minimal monic polynomial of w over Fp factors into those r factors in Fp(w)?

Here if w is replaced with a primitive element of Fp(w) then it's easy to prove. But w is not a primitive element of Fp(w), it's a primitive element of Fp(w)/Fp.

Suppose w is a primitive n-th root of unity, let r = Ord_n(p) (the smallest integer s.t. $p^r=1 mod n $), $f(w)=w^p$(the frobenius map), how to prove that $w,f(w),f^2(w),...f^{r-1}(w)$ are all different in Fp(w) so that the minimal monic polynomial of w over Fp factors into those r factors in Fp(w)?

Here if w is replaced with a primitive element of Fp(w) then it's easy to prove. But w is not a primitive element of Fp(w), it's a primitive element of Fp(w)/Fp.

Suppose w is a primitive n-th root of unity, let r = Ord_n(p), f(w)=w^p(the frobenius map), how to prove that w,f(w),f^2(w),...f^(r-1)(w) are all different in Fp(w) so that the minimal monic polynomial of w over Fp factors into those r factors in Fp(w)?

Here if w is replaced with a primitive element of Fp(w) then it's easy to prove. But w is not a primitive element of Fp(w), it's a primitive element of Fp(w)/Fp.

Suppose w is a primitive n-th root of unity, let r = Ord_n(p), $f(w)=w^p$(the frobenius map), how to prove that $w,f(w),f^2(w),...f^{r-1}(w)$ are all different in Fp(w) so that the minimal monic polynomial of w over Fp factors into those r factors in Fp(w)?

Here if w is replaced with a primitive element of Fp(w) then it's easy to prove. But w is not a primitive element of Fp(w), it's a primitive element of Fp(w)/Fp.