Algebraic number theory MOC

Number field

A number field is an extension field of of finite degree , #m/def/num/alg whence is an algebraic extension. Similarly, if is an arbitrary extension and is algebraic over , then is a number field,1 and is called an algebraic number.2

Sage

To create a number field in Sage with a given defining polynomial,

f = x^3 - 2 * x - 2
K.<θ> = NumberField(f)

In order to study a number field we often turn to study its ring of integers and ideal class group.

Properties

Classification

By degree

By form

By properties

#state/tidy | #lang/en | #SemBr

Footnotes

  1. All number fields have this form by the Primitive element theorem. ↩

  2. 2022. Algebraic number theory course notes. §2, p. 7 ↩