Definition
Number field
A finite field extension of the rational numbers.
Definition
A number field is a finite field extension
of the rational numbers. Its degree is the finite dimension .
Places and completions
The embeddings of into and determine its archimedean places. Its remaining places are nonarchimedean and lie above rational primes. Completing at any place produces a local field.
Position among global fields
Number fields are precisely the global fields of characteristic . The other global fields are the global function fields, which have positive characteristic.
References
- Jürgen Neukirch, Algebraic Number Theory, Springer, 1999, Chapter I.
- John W. S. Cassels and Albrecht Fröhlich, eds., Algebraic Number Theory, Academic Press, 1967.