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If $K$ has characteristic $p>0$, an Artin-Schreier extension $L/K$ is cyclic of degree $p$, and is generated by an element $a \in L$ such that $a^p \not \in K$. Such an $L$ has the form $K(a)$ for a root $a$ of a polynomial of the form $X^p - X - \alpha$ for suitable $\alpha \in K$; the polynomial then splits over $L$ since its roots are $a+i$ for $i \in \mathbf{F}_p$ (in this way you can see the action of the Galois group $\mathbf{Z}/p$ on the roots).

An example of such an extension is $\mathbf{F}_4/\mathbf{F}_2$ since $\mathbf{F}_4 = \mathbf{F}_2(a)$ for a root $a$ of $X^2 - X - 1 = X^2 + X +1$.

Edit(x2): Looking at Georges Elencwajg's answer, I see that I made what he labels the "First Interpretation" of the question. Since his "Second Interpetation" seems interesting, here is what I hope is an example.

Let $k$ be a field of char $p>0$, and let $K = k(t)$ for transcendental $t$. Let $L/K$ be the Artin-Schreier extension obtained by adjoining to $K$ a root $a$ of $X^p - X - t$.

I claim that $a^N \not \in K$ for any $N \ge 1$.

Well, note that $k[a,t]$ is an integral extension of $k[t]$. Hence if $a^N \in K$, then in fact $a^N \in \langle t \rangle \subset k[t]$, so that $a$ satisfies a polynomial $$X^N - tg(t) \quad \text{for some $g(t) \in k[t]$}.$$ But then $X^p - X - t$ divides $X^N - tg(t)$ and reducing mod $t$ gives a contradiction. (i.e. the condition $a^N \in K$ would mean that $L/K$ should be ramified at the prime $\langle t \rangle$ of $k[t]$, which isn't so).

So: this cyclic extension is generated by an element $a$ for which $a^N \not \in K$ for any $N \ge 1$. I expect (?!) that this should give an example for Elencwajg's "Second Interpretation" of the question, but I didn't think about other possible generators for the extension $L/K$.

If $K$ has characteristic $p>0$, an Artin-Schreier extension $L/K$ is cyclic of degree $p$, and is generated by an element $a \in L$ such that $a^p \not \in K$. Such an $L$ has the form $K(a)$ for a root $a$ of a polynomial of the form $X^p - X - \alpha$ for suitable $\alpha \in K$; the polynomial then splits over $L$ since its roots are $a+i$ for $i \in \mathbf{F}_p$ (in this way you can see the action of the Galois group $\mathbf{Z}/p$ on the roots).

An example of such an extension is $\mathbf{F}_4/\mathbf{F}_2$ since $\mathbf{F}_4 = \mathbf{F}_2(a)$ for a root $a$ of $X^2 - X - 1 = X^2 + X +1$.

Edit(x2): Looking at Georges Elencwajg's answer, I see that I made what he labels the "First Interpretation" of the question. Since his "Second Interpetation" seems interesting, here is what I hope is an example.

Let $k$ be a field of char $p>0$, and let $K = k(t)$ for transcendental $t$. Let $L/K$ be the Artin-Schreier extension obtained by adjoining to $K$ a root $a$ of $X^p - X - t$.

I claim that $a^N \not \in K$ for any $N \ge 1$.

Well, note that $k[a,t]$ is an integral extension of $k[t]$. Hence if $a^N \in K$, then in fact $a^N \in \langle t \rangle \subset k[t]$, so that $a$ satisfies a polynomial $$X^N - tg(t) \quad \text{for some $g(t) \in k[t]$}.$$ But then $X^p - X - t$ divides $X^N - tg(t)$ and reducing mod $t$ gives a contradiction. (i.e. the condition $a^N \in K$ would mean that $L/K$ should be ramified at the prime $\langle t \rangle$ of $k[t]$, which isn't so).

So: this cyclic extension is generated by an element $a$ for which $a^N \not \in K$ for any $N \ge 1$. I expect (?!) that this should give an example for Elencwajg's "Second Interpretation" of the question, but I didn't think about other possible generators for the extension $L/K$.

If $K$ has characteristic $p>0$, an Artin-Schreier extension $L/K$ is cyclic of degree $p$, and is generated by an element $a \in L$ such that $a^p \not \in K$. Such an $L$ has the form $K(a)$ for a root $a$ of a polynomial of the form $X^p - X - \alpha$ for suitable $\alpha \in K$; the polynomial then splits over $L$ since its roots are $a+i$ for $i \in \mathbf{F}_p$ (in this way you can see the action of the Galois group $\mathbf{Z}/p$ on the roots).

An example of such an extension is $\mathbf{F}_4/\mathbf{F}_2$ since $\mathbf{F}_4 = \mathbf{F}_2(a)$ for a root $a$ of $X^2 - X - 1 = X^2 + X +1$.

Edit(x2): Looking at Georges Elencwajg's answer, I see that I made what he labels the "First Interpretation" of the question. Since his "Second Interpetation" seems interesting, here is an example.

Let $k$ be an algebraic closure of $\mathbf{F}_p$, and let $K = k(t)$ for transcendental $t$. Let $L/K$ be the Artin-Schreier extension obtained by adjoining to $K$ a root $a$ of $X^p - X - t$.

I claim that $a^N \not \in K$ for any $N \ge 1$.

Well, note that $k[a,t]$ is an integral extension of $k[t]$. Hence if $a^N \in K$, then in fact $a^N \in \langle t \rangle \subset k[t]$, so that $a$ satisfies a polynomial $$X^N - tg(t) \quad \text{for some $g(t) \in k[t]$}.$$ But then $X^p - X - t$ divides $X^N - tg(t)$ and reducing mod $t$ gives a contradiction. (i.e. the condition $a^N \in K$ would mean that $L/K$ should be ramified at the prime $\langle t \rangle$ of $k[t]$, which isn't so).

So: this cyclic extension is generated by an element $a$ for which $a^N \not \in K$ for any $N \ge 1$. I expect (?!) that this should give an example for Elencwajg's "Second Interpretation" of the question, but I didn't think about other possible generators for the extension $L/K$.

If $K$ has characteristic $p>0$, an Artin-Schreier extension $L/K$ is cyclic of degree $p$, and is generated by an element $a \in L$ such that $a^p \not \in K$. Such an $L$ has the form $K(a)$ for a root $a$ of a polynomial of the form $X^p - X - \alpha$ for suitable $\alpha \in K$; the polynomial then splits over $L$ since its roots are $a+i$ for $i \in \mathbf{F}_p$ (in this way you can see the action of the Galois group $\mathbf{Z}/p$ on the roots).

An example of such an extension is $\mathbf{F}_4/\mathbf{F}_2$ since $\mathbf{F}_4 = \mathbf{F}_2(a)$ for a root $a$ of $X^2 - X - 1 = X^2 + X +1$.

Edit(x2): Looking at Georges Elencwajg's answer, I see that I made what he labels the "First Interpretation" of the question. Since his "Second Interpetation" seems interesting, here is what I hope is an example.

Let $k$ be a field of char $p>0$, and let $K = k(t)$ for transcendental $t$. Let $L/K$ be the Artin-Schreier extension obtained by adjoining to $K$ a root $a$ of $X^p - X - t$.

I claim that $a^N \not \in K$ for any $N \ge 1$.

Well, note that $k[a,t]$ is an integral extension of $k[t]$. Hence if $a^N \in K$, then in fact $a^N \in \langle t \rangle \subset k[t]$, so that $a$ satisfies a polynomial $$X^N - tg(t) \quad \text{for some $g(t) \in k[t]$}.$$ But then $X^p - X - t$ divides $X^N - tg(t)$ and reducing mod $t$ gives a contradiction. (i.e. the condition $a^N \in K$ would mean that $L/K$ should be ramified at the prime $\langle t \rangle$ of $k[t]$, which isn't so).

So: this cyclic extension is generated by an element $a$ for which $a^N \not \in K$ for any $N \ge 1$. I expect (?!) that this should give an example for Elencwajg's "Second Interpretation" of the question, but I didn't think about other possible generators for the extension $L/K$.

If $K$ has characteristic $p>0$, an Artin-Schreier extension $L/K$ is cyclic of degree $p$, and not of the indicated form. Such an $L$ has the form $K(a)$ for a root $a$ of a polynomial of the form $X^p - X - \alpha$ for a suitable $\alpha \in K$; the polynomial then splits over $L$ since its roots are all $a+i$ for $i \in \mathbf{F}_p$ (in this way you can see the action of the Galois group $\mathbf{Z}/p$ on the roots).

An example of such an extension is $\mathbf{F}_4/\mathbf{F}_2$ since $\mathbf{F}_4 = \mathbf{F}_2(a)$ for a root $a$ of $X^2 - X - 1 = X^2 + X +1$.

Edit: whoops. Looking at Georges Elencwajg answer, I see that I made what he labels as the "First Interpretation" of the question (which was actually a bit sloppy of me...).

If $K=\mathbf{F}_p(t)$ for transcendental $t$, and if $L/K$ is the Artin-Schreier extension obtained by adjoining to $K$ a root a of $X^p - X - t$, then for $i\ge 1$, $a^{p^i} =a+f(t)$ where $f(t) \in \mathbf{F}_p[t]$ is a polynomial of degree $p^{i-1}$. This show that $a^N \not \in K$ for any $N\ge 1$. So this cyclic extension gives an example meeting Elencwajg's "Second Interpretation" of the question.

If $K$ has characteristic $p>0$, an Artin-Schreier extension $L/K$ is cyclic of degree $p$, and is generated by an element $a \in L$ such that $a^p \not \in K$. Such an $L$ has the form $K(a)$ for a root $a$ of a polynomial of the form $X^p - X - \alpha$ for suitable $\alpha \in K$; the polynomial then splits over $L$ since its roots are $a+i$ for $i \in \mathbf{F}_p$ (in this way you can see the action of the Galois group $\mathbf{Z}/p$ on the roots).

An example of such an extension is $\mathbf{F}_4/\mathbf{F}_2$ since $\mathbf{F}_4 = \mathbf{F}_2(a)$ for a root $a$ of $X^2 - X - 1 = X^2 + X +1$.

Edit(x2): Looking at Georges Elencwajg's answer, I see that I made what he labels the "First Interpretation" of the question. Since his "Second Interpetation" seems interesting, here is an example.

Let $k$ be an algebraic closure of $\mathbf{F}_p$, and let $K = k(t)$ for transcendental $t$. Let $L/K$ be the Artin-Schreier extension obtained by adjoining to $K$ a root $a$ of $X^p - X - t$.

I claim that $a^N \not \in K$ for any $N \ge 1$.

Well, note that $k[a,t]$ is an integral extension of $k[t]$. Hence if $a^N \in K$, then in fact $a^N \in \langle t \rangle \subset k[t]$, so that $a$ satisfies a polynomial $$X^N - tg(t) \quad \text{for some $g(t) \in k[t]$}.$$ But then $X^p - X - t$ divides $X^N - tg(t)$ and reducing mod $t$ gives a contradiction. (i.e. the condition $a^N \in K$ would mean that $L/K$ should be ramified at the prime $\langle t \rangle$ of $k[t]$, which isn't so). 

So: this cyclic extension is generated by an element $a$ for which $a^N \not \in K$ for any $N \ge 1$. I expect (?!) that this should give an example for Elencwajg's "Second Interpretation" of the question, but I didn't think about other possible generators for the extension $L/K$.