View source for Tower of fields


{{MSC|12Fxx|11R37}}
{{TEX|done}}

A tower of fields or a 
field tower is
an extension sequence 
$$k\subset k_1\subset \dots \subset k_i \subset \dots$$
of some field $k$. Depending on the
properties of the extensions $k_{i+1}/k_i$, the tower is called normal, Abelian,
separable, etc. The concept of a field tower plays an important role
in
[[Galois theory|Galois theory]], in which the problem of expressing
the roots of equations by radicals is reduced to the possibility of
including the splitting field of the equation into a normal Abelian
field tower.

In
[[Class field theory|class field theory]] the tower 
$$k\subset k_1\subset \dots \subset k_i \subset \dots$$
occurs,
where $k$ is some algebraic number field, while each field $k_{i+1}$ is the
Hilbert class field of $k_i$ (i.e. the maximal Abelian unramified
extension of $k_i$). The Galois group of any extension $k_{i+1}/k_i$ is isomorphic
to the ideal class group of the field $k_i$ (by Artin's reciprocity law)
and, since the latter group is finite, all extensions $k_{i+1}/k_i$ are finite
as well. The union $K$ of the fields $k_i$ is the maximal solvable
unramified extension of $k$. The question of the finiteness of the
extension $K/k$ (the class field tower problem) was posed in 1925 by
Ph. Furtwängler and was negatively answered in 1964
{{Cite|GoSh}}. An example of a field with an infinite class
field tower is the extension of the field of rational numbers obtained
by adjoining $\sqrt{-30030}$. It is impossible to imbed such a field in an
algebraic number field that has unique factorization. The solution of
the problem has applications in algebraic number theory, e.g. in
obtaining a precise estimate of the growth of discriminants of
algebraic number fields.

====References====
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|valign="top"|{{Ref|CaFr}}||valign="top"| J.W.S. Cassels (ed.)  A. Fröhlich (ed.), ''Algebraic number theory'', Acad. Press (1967) {{MR|0215665}} {{ZBL|0153.07403}}
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|valign="top"|{{Ref|GoSh}}||valign="top"| E.S. Golod, I.R. Shafarevich, "On class field towers" ''Transl. Amer. Math. Soc. (2)'', '''48''' (1965) pp. 91–102 ''Izv. Akad. Nauk SSSR Ser. Mat.'', '''28''' (1964) pp. 261–272 {{MR|0161852}} {{ZBL|0148.28101}}
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