about the totally ramified subextension - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T19:15:56Z http://mathoverflow.net/feeds/question/88078 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88078/about-the-totally-ramified-subextension about the totally ramified subextension unknown (yahoo) 2012-02-10T07:11:40Z 2012-02-10T07:11:40Z <p>For any Galois extension of local field $L/K$, it's easy to find the maximal unramified sub-extension $K^{ur}\cap L$ of $K$. But is there any method to find the totally ramified sub-extension of $K$? Is there any such extension related to the totally ramified extension $L/K^{ur}\cap L$? </p> <p>I try to find an Eisenstein polynomial, and then adjoin a root $\alpha$ to $K$ to find such totally ramified extension. But I cannot modify the polynomial $x^e-\pi_L^e$ to let it inside $K[x]$. Should I consider in this way?</p>