Tower of fields

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

A tower of fields or a field tower is an extension sequence $$k\subset k_1\subset \cdots \subset k_i\subset \dots$$ 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 $$k\subset k_1\subset \cdots \subset k_i\subset \dots$$ 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.

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