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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 )

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    $\begingroup$ 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? $\endgroup$ Dec 10, 2014 at 18:59
  • $\begingroup$ Yes.you are right, sir $\endgroup$ Dec 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. $\endgroup$ Dec 10, 2014 at 19:17
  • $\begingroup$ A lot of examples of this kind are given in Cox, "Primes of the form $x^2 + ny^2$". $\endgroup$
    – fretty
    Dec 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. $\endgroup$ Dec 11, 2014 at 13:19

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