Definition:Number Field
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Definition
A number field is a finite extension of the Field of Rational Numbers $\Q$.
Alternatively, a number field can be defined as a subfield of the Field of Complex Numbers $\C$ with finite degree over $\Q$.
Notes
The first definition is more inclusive than the second.
For example, $\Q[x]/(x^2-2)$ is a finite extension of $\Q$ but is not a subfield of $\C$.
Also, in the second definition, it is important that the subfield have finite degree over $\Q$; for example, $\Q \subset \R \subset \C$ but $\R$ is not a number field since $[\R:\Q] = \infty$.
Sources
- C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 4.17$
