Algebraic number theory MOC

Number field

A number field is an extension field of Rational numbers of finite degree , alg whence is an algebraic extension. Similarly, if is an arbitrary extension and is algebraic over , then [[Adjunction of a ring|]] 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

• Bases for a number field
• Norm of a number field

Classification

By degree

• Quadratic field
• Cubic field

By form

• Cyclotomic field

By properties

• Monogenic field

---

tidy | 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 ↩