Concept

Tower of fields

In mathematics, a tower of fields is a sequence of field extensions F0 ⊆ F1 ⊆ ... ⊆ Fn ⊆ ... The name comes from such sequences often being written in the form A tower of fields may be finite or infinite. Q ⊆ R ⊆ C is a finite tower with rational, real and complex numbers. The sequence obtained by letting F0 be the rational numbers Q, and letting (i.e. Fn+1 is obtained from Fn by adjoining a 2n th root of 2) is an infinite tower. If p is a prime number the p th cyclotomic tower of Q is obtained by letting F0 = Q and Fn be the field obtained by adjoining to Q the pn th roots of unity. This tower is of fundamental importance in Iwasawa theory. The Golod–Shafarevich theorem shows that there are infinite towers obtained by iterating the Hilbert class field construction to a number field.

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https://en.wikipedia.org/wiki/Tower_of_fields

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Algebraic number field

In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic field extension). Thus is a field that contains and has finite dimension when considered as a vector space over . The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

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