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Kummer Theory and Reciprocity Laws
Peter Stevenhagen 2002
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2. Kummer Theory
If L/K is a cyclic extension of degree n but ζn not∈ K, it can be very hard to describe.
However, when one adjoins ζn to K, by Kummer theory,
the extension L(ζn) is now Kummer over K(ζn), so L = K(ζn,a^(1/n)) for some a ∈ K(ζn).
Note that this extension is not abelian, but it is abelian in two steps, which is usually good enough.
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Galois theory for schemes
H. W. Lenstra
Mathematisch Instituut
Universiteit Leiden
Electronic third edition: 2008
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Introduction 1?5
Coverings of topological spaces. The fundamental group. Finite ´etale coverings of a scheme.
An example. Contents of the sections. Prerequisites and conventions.
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Algebra course notes