Let $K$ be the algebraic closure of $\mathbf{Z}/p\mathbf{Z}$,and let $w$ belongs to $Gal(K/(\mathbf{Z}/p\mathbf{Z}))$ be the Frobenius map sending $a$ to $a^p$.I have shown that $w$ has infinite order. I want an example of a map in $Gal(K/(\mathbf{Z}/p\mathbf{Z}))$ which does not belong to the cyclic subgroup generated by $w$.
Algebraic closure of $\mathbf{Z}/p\mathbf{Z}$
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galois-theory
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1The Galois group is (isomorphic to) $\hat{\mathbb{Z}}$, so just pick your favorite element of $\hat{\mathbb{Z}}$ that is not contained in $\mathbb{Z}$, and that will give you the map you seek. If you aren't familiar with $\hat{\mathbb{Z}}$, then you should look into it! One keyword is "procyclic group". – 2012-04-19
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0A [related question.](http://math.stackexchange.com/q/27119/11619) – 2012-04-19