mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a tower of fields is a sequence of

field extension In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...

s : The name comes from such sequences often being written in the form :

\begin\vdots \\ ,  \\ F_2 \\ ,  \\ F_1 \\ ,  \\ \ F_0. \end

A tower of fields may be finite or infinite.

Examples

* is a finite tower with

rational Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...

, real and

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s. *The sequence obtained by letting ''F''₀ be the rational numbers Q, and letting ::

F_ = F_\!\left(2^\right), \quad \text\ n \geq 1

:(i.e. ''F''_''n'' is obtained from ''F''_''n-1'' by adjoining a 2^''n''th

root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...

of 2), is an infinite tower. *If ''p'' is a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

the ''p''th

cyclotomic In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory b ...

tower of Q is obtained by letting ''F''₀ = Q and ''F''_''n'' be the field obtained by adjoining to Q the ''pⁿ''th roots of unity. This tower is of fundamental importance in

Iwasawa theory In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite Tower of fields, towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic ...

. *The Golod–Shafarevich theorem shows that there are infinite towers obtained by iterating the

Hilbert class field In algebraic number theory, the Hilbert class field ''E'' of a number field ''K'' is the Maximal abelian extension, maximal abelian unramified extension of ''K''. Its degree over ''K'' equals the class number of ''K'' and the Galois group of ''E'' ...

construction to a

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

References

*Section 4.1.4 of {{Citation , last=Escofier , first=Jean-Pierre , title=Galois theory , publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, series=

Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) () is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with va ...

, volume=204 , year=2001 , isbn=978-0-387-98765-1 Field extensions