Algebraic number theory MOC

Number field

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

By degree

• Quadratic field
• Cubic field

By form

• Cyclotomic field

By properties

• Monogenic field

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Footnotes

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

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