Tower of Fields

Tower of Fields - a sequence of extensions for some field $K$ ${\ displaystyle K}$ $K$ : $K\subset K_{1}\subset \dots \subset K_{i}\subset \dots$ ${\ displaystyle K \ subset K_ {1} \ subset \ dots \ subset K_ {i} \ subset \ dots}$ $K \ subset K_ {1} \ subset \ dots \ subset K_ {i} \ subset \ dots$ may be finite or infinite. Often recorded vertically:

{\begin{array}{c}\vdots \\|\\K_{i}\\|\\\vdots \\|\\K_{1}\\|\\K\end{array}}

{\ displaystyle {\ begin {array} {c} \ vdots \\ | \\ K_ {i} \\ | \\\ vdots \\ | \\ K_ {1} \\ | \\ K \ end {array} }}

{\ begin {array} {c} \ vdots \\ | \\ K_ {i} \\ | \\\ vdots \\ | \\ K_ {1} \\ | \\ K \ end {array}}

For example, $\mathbb {Q} \subset \mathbb {R} \subset \mathbb {C}$ ${\ displaystyle \ mathbb {Q} \ subset \ mathbb {R} \ subset \ mathbb {C}}$ ${\ displaystyle \ mathbb {Q} \ subset \ mathbb {R} \ subset \ mathbb {C}}$ - the ultimate tower of extensions of the field of rational numbers , successively including the fields of real and complex numbers.

A normal tower of fields is a sequence of normal extensions , a separable tower of fields is a sequence of separable extensions , an Abelian tower of fields is a sequence of Abelian extensions .

The classical problem of solvability in the radicals of polynomials, solved by means of Galois theory , can be formulated in terms of field towers: solvability is equivalent to the embeddability of the coefficient field of a given polynomial normal and an abelian tower of fields.

The class field tower is a field tower built over a certain field of algebraic numbers , each element of which is a maximal Abelian unramified extension of the previous one. One of the results of class field theory , which entails important consequences for algebraic number theory, is a negative solution to the unbounded Burnside problem ( the Golod – Shafarevich theorem ), in the language of class fields is formulated as follows: there are infinite class field towers ^[1] ^[2] (in particular, such is the tower built over the extension of the field of rational numbers obtained by adding the number ${\sqrt {30030}}$ ${\ displaystyle {\ sqrt {30030}}}$ ${\ sqrt {30030}}$ ).

Notes

↑ Golod E. S. On nil-algebras and finitely approximable p-groups // Izvestiya AN SSSR. Mathematical series. - 1964. - V. 28, issue 2 . - p . 273-276 .
↑ Golod E.S. , Shafarevich I.R. About the class field tower // News of the Academy of Sciences of the USSR. Mathematical series. - 1964. - V. 28, issue 2 . - p . 261-272 .

Literature

Tower of the Fields - an article from the Mathematical Encyclopedia . A. N. Parshin

[1] Golod E. S. On nil-algebras and finitely approximable p-groups // Izvestiya AN SSSR. Mathematical series. - 1964. - V. 28, issue 2 . - p . 273-276 .

[2] Golod E.S. , Shafarevich I.R. About the class field tower // News of the Academy of Sciences of the USSR. Mathematical series. - 1964. - V. 28, issue 2 . - p . 261-272 .

Tower of Fields

Notes

Literature

More articles: