Suppose that $D$ is a division ring with center $F$ and with index $p$
prove that $D$ is cyclic if and only if there exists $x$ $\notin$$F$ which $x$$^p$ $\in$$F$.
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2Can you add the definition of cyclic? – egreg Jan 23 '16 at 18:13
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One direction follows from the definition of cyclicity (at least from the version I'm familiar with - you get $x$ from Skolem-Noether and double centralizer theorems). For the other you need to somehow produce a cyclic extension of $F$ as a subfield. What toold have you got at your disposal? – Jyrki Lahtonen Jan 25 '16 at 11:50
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Oh, and welcome to Math.SE! You are asking interesting questions. Just to be on the safe side I want to inform you that questions are received better here, if you include relevant context. Definitions, results known to you, your own thoughts and any partial work you have done. Also, we don't have too many people here who are experts on division rings, so you brace yourself for a waiting period :-) – Jyrki Lahtonen Jan 25 '16 at 11:53