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''field tower''
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{{TEX|done}}
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{{MSC|12Fxx|11R37}}
  
An extension sequence
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A tower of fields or a
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field tower is
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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,
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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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t0935301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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In
 
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[[Class field theory|class field theory]] the tower  
of some field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t0935302.png" />. Depending on the properties of the extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t0935303.png" />, 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.
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$$k\subset k_1\subset \dots \subset k_i \subset \dots$$
 
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occurs,
In [[Class field theory|class field theory]] the tower
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where $k$ is some algebraic number field, while each field $k_{i+1}$ is the
 
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Hilbert class field of $k_i$ (i.e. the maximal Abelian unramified
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t0935304.png" /></td> </tr></table>
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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)
occurs, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t0935305.png" /> is some algebraic number field, while each field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t0935306.png" /> is the Hilbert class field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t0935307.png" /> (i.e. the maximal Abelian unramified extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t0935308.png" />). The Galois group of any extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t0935309.png" /> is isomorphic to the ideal class group of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t09353010.png" /> (by Artin's reciprocity law) and, since the latter group is finite, all extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t09353011.png" /> are finite as well. The union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t09353012.png" /> of the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t09353013.png" /> is the maximal solvable unramified extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t09353014.png" />. The question of the finiteness of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t09353015.png" /> (the class field tower problem) was posed in 1925 by Ph. Furtwängler and was negatively answered in 1964 [[#References|[2]]]. An example of a field with an infinite class field tower is the extension of the field of rational numbers obtained by adjoining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093530/t09353016.png" />. 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.
+
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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD 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</TD></TR></table>
+
{|
 +
|-
 +
|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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|}

Revision as of 23:28, 17 February 2012

2020 Mathematics Subject Classification: Primary: 12Fxx Secondary: 11R37 [MSN][ZBL]

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, 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 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 [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

[CaFr] J.W.S. Cassels (ed.) A. Fröhlich (ed.), Algebraic number theory, Acad. Press (1967) MR0215665 Zbl 0153.07403
[GoSh] 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 MR0161852 Zbl 0148.28101
How to Cite This Entry:
Tower of fields. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tower_of_fields&oldid=16624
This article was adapted from an original article by A.N. Parshin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article