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Let K be a field of characteristic p.

Suppose f(x)=(x

My question is:

1. since f(x) in K[x], can I conclude g(x)=x

2. We see that in general if g(x)=x

Now if c

very appreciated if someone could solve my confusion.

Thank you!

Suppose f(x)=(x

^{k}+c_{k-1}x^{k-1}+...+c_{0})(x^{p-k}+.........) in K[x] with 1≤k≤p-1.My question is:

1. since f(x) in K[x], can I conclude g(x)=x

^{k}+c_{k-1}x^{k}+...+c_{0}in K[x] as well?2. We see that in general if g(x)=x

^{k}+c_{k-1}x^{k-1}+...+c_{0}then c_{k-1}=-(α_{1}+α_{2}+...+α_{k}) where α_{1},α_{1},...,α_{k}are the root of g(x).Now if c

_{k-1}∈K, can I conclude that α_{1},α_{1},...,α_{k}∈K as well?very appreciated if someone could solve my confusion.

Thank you!

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