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Ramification Groups of Nonabelian Kummer Extensions

✍ Scribed by Romyar T Sharifi

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
329 KB
Volume
65
Category
Article
ISSN
0022-314X

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✦ Synopsis

The reciprocity law of Coleman for the Hilbert norm residue symbol has allowed the computation of the conductors of the abelian Kummer extensions Q p ( p n a, pn )Γ‚Q p (pn ) with a # Q p and `pn a primitive ( p n ) th root of unity for a fixed prime p and all positive integers n. From these conductors, we compute the ramification groups of the nonabelian Kummer extension Q p ( p -Q _ p )Γ‚Q p obtained from adjoining to Q p all p-power roots of its elements. More generally, given a similar nonabelian Kummer extension of complete discrete valuation fields, we have a method of computing its ramification groups from the conductors of the abelian Kummer extensions and knowledge of the ramification groups of the cyclotomic extensions.

1997 Academic Press Coleman in [1], we shall determine the ramification groups of the extensions Q p ( p n -Q _ p )Γ‚Q p .

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