Imaginary quadratic field

ℚ(√−21)

Consider the monogenic imaginary quadratic field 𝐾 =ℚ(𝛼) where 𝛼 =√−21. #m/thm/num/alg

K.<α> = QuadraticField(-21)

Discriminant

By Discriminant of an algebraic integer,

Group of units

By ^P1,

Class group

Minkowski's bound is given by

so applying Kummer's factorization theorem

Clearly no algebraic integers can have these norms, so we can be satisfied that these are not principal. Since 𝔭−15 ∼𝔭′5, the ideal class group is generated by {[𝔭1],[𝔭2],[𝔭3]}. Some algebraic integers of small field norm are

whence

• from 𝑡 =2 we see 𝔭25 =⟨5,𝛼 +2⟩2 =⟨𝛼 +2⟩ ∼⟨1⟩;
• from 𝑡 =3 we see 𝔭2𝔭3𝔭5 =⟨𝛼 −3⟩ ∼⟨1⟩.

so we see Cl⁡𝐾 ≅V4.

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