Recently J. Coates and R. Greenberg introduced a new important
class of extensions of local number
fields with finite residue field which they call deeply ramified fields.
These extensions play an essential role
in their study of the arithmetics of abelian varieties over local fields
with finite residue fields [1].
The first aim of this paper is to provide an ‘elementary’
treatment of deeply ramified extensions of local
fields with arbitrary perfect residue fields using a method different from
the original approach of Coates
and Greenberg. Equivalent properties (1), (3)–(8) of deeply ramified
extensions in Section 1 are due to
them, and for their proofs a presentation of the different as an integral
was involved. We translate the most
important constructions into the language of the Hasse–Herbrand function
(Section 1) and then apply the
methods of [3, Chapter 3], where the Hasse–Herbrand
function is defined in terms of the norm map. Some
of the properties of deeply ramified extensions (two implications in the
language of this paper) have been
already studied by M. Matignon [7] and J. Fresnel, M.
Matignon [5] for different purposes (see Remark 1.6
and the beginning of Section 2).
The second aim of this text is to expose relations among classes of
deeply ramified extensions, that of
arithmetically profinite extensions (Subsections 2.1–2.4) and that
of p-adic Lie extensions (Subsections
2.5–2.7). For local fields with finite residue field we give in Subsection
2.2 an example of a Galois deeply
ramified extension with infinite residue extension in which every Galois
deeply ramified subextension is not
arithmetically profinite; and in Subsection 2.4 we give an example of a
Galois deeply ramified extension
with finite residue field extension and a nondiscrete set of breaks (that
means that this extension is not
arithmetically profinite). The main result is that for local fields with
finite residue field the class of Galois
deeply ramified extensions with finite residue extension and a discrete
set of breaks coincides with the class
of infinite Galois arithmetically profinite extensions (Proposition 2.3).
However, in the case of the fields
with infinite residue fields Proposition 2.3 does not hold (Subsection
2.1). In Subsection 2.5 we construct
an example that shows that the class of infinite Galois totally ramified
arithmetically profinite extensions
is strictly larger than the class of the most natural arithmetic origin
– the class of totally ramified p-adic Lie
extensions. Subsection 2.6 contains an example of a Galois deeply and totally
ramified extension L of a
local field F with finite residue field such that the norm group
of L/E is of finite index in E*.