about Kummer Theory - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T09:19:10Z http://mathoverflow.net/feeds/question/88242 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88242/about-kummer-theory about Kummer Theory unknown (google) 2012-02-11T22:57:20Z 2012-04-08T11:22:00Z <p>In Kummer Theory, we always need the assumption that $a\not\in K^{\times n}$. For the local case, we can use the structure theorem of $K^{\times}$ to check this assumption. But in general, how can we check this condition?</p> <p>For example, how to check $p\not\in K^{\times p}$, where $K=\mathbb{Q}(\mu_{p^\infty})$ the rational number adjoining all the pth power roots of unity.</p> http://mathoverflow.net/questions/88242/about-kummer-theory/90893#90893 Answer by Tommaso Centeleghe for about Kummer Theory Tommaso Centeleghe 2012-03-11T09:55:06Z 2012-03-11T10:12:01Z <p>In</p> <p><a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.jmsj/1261734945" rel="nofollow">http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.jmsj/1261734945</a></p> <p>Chevalley shows the following related statement (Remarque p. 39): let $p$ be a prime, $K$ a field of characteristic different from $p$, and $\zeta$ a $p^e$-th primitive root of $1$ in some algebraic extension of $K$, where $e\geq 1$ is any integer. Assume moreover that $-1$ is a square in $K$ if $p=2$. Then if an element $x\in K(\zeta)$ is a $p^e$-th power, then it is already a $p^e$-th power in $K$. </p>