algebraic number field

English

Noun

algebraic number field (plural algebraic number fields)

1. (mathematics, algebraic number theory) A field which includes the rational numbers and has finite dimension as a vector space over the rational numbers.

   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.

   • 1984, Alan Baker, A Concise Introduction to the Theory of Numbers, Cambridge University Press, page 61:

     Although we shall be concerned in the sequel only with quadratic fields, we shall nevertheless begin with a short discussion of the more general concept of an algebraic number field.

   • 1989, M. Pohst, H. Zassenhaus, Algorithmic Algebraic Number Theory, 1st paperback edition, Cambridge University Press, published 1997, page 381:

     In this section we consider the maximal order of an algebraic number field and discuss the most important consequences which follow from the fact of it being a Dedekind ring.

   • 2006, Andres Iglesias, Nobuki Takayama (editors), Mathematical Software - ICMS 2006: 2nd International Congress, Proceedings, Masayuki Noro, An Efficient Implementation for Computing Gröbner Bases over Algebraic Number Fields, Springer, LNCS 4151, page 99,

     If the operations over an algebraic number field are provided, we can apply [the] usual Buchberger algorithm over the field.

Synonyms

• number field

Hypernyms

• associative algebra
• extension field

Hyponyms

• cyclotomic field
• quadratic field

Meronyms

• number ring
• ring of integers

Further reading

• Algebraic number ring on Wikipedia.Wikipedia
• Algebraic number theory on Wikipedia.Wikipedia