Can anyone explain with a numerical example of generating class field with Kummer extension? I have not come across any standard reference which does give examples. Please help or cite any reference for the same.We assume that the base field K contains roots of unity. (Artin -Tate,Milne,Lang,Childress,Cohen etc.consulted )
$\begingroup$
$\endgroup$
6
-
1$\begingroup$ Can you be a bit more precise? Are you looking for a number field $F$ such that its Hilbert class field $H_F$ can be written as $H_F=F(\sqrt[n]{\alpha})$ for some $n$ and $\alpha$? Or am I misunderstanding? $\endgroup$– Filippo Alberto EdoardoDec 10, 2014 at 18:59
-
$\begingroup$ Yes.you are right, sir $\endgroup$– RAJRATNA AdsulDec 10, 2014 at 19:05
-
$\begingroup$ But explicitly with some value of $\alpha$ and n and discerning the entire process in action to finally generate $H_F$ . Incidentally, the way the Existence Theorem works. $\endgroup$– RAJRATNA AdsulDec 10, 2014 at 19:17
-
$\begingroup$ A lot of examples of this kind are given in Cox, "Primes of the form $x^2 + ny^2$". $\endgroup$– frettyDec 10, 2014 at 21:07
-
$\begingroup$ Daniel Fretwell sir,I have found your exposition of class field theory the coolest one,wherein you have discussed later portions based on Cox,and Childress. $\endgroup$– RAJRATNA AdsulDec 11, 2014 at 13:19
|
Show 1 more comment