Number field

A number field $𝐾$ is an extension field of $ℚ$ of finite degree $[𝐾 : ℚ]$ , #m/def/num/alg whence $𝐾 : ℚ$ is an algebraic extension. Similarly, if $𝐾 : ℚ$ is an arbitrary extension and $𝑥 \in 𝐾$ is algebraic over $ℚ$ , then $ℚ (𝑥)$ is a number field,¹ and $𝑥$ 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

Bases for a number field
Norm of a number field

Classification

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

All number fields have this form by the Primitive element theorem. ↩
2022. Algebraic number theory course notes. §2, p. 7 ↩

Number field

Properties

Classification

By degree

By form

By properties

Footnotes