Imaginary quadratic field

ℚ(√−21)

Consider the monogenic imaginary quadratic field 𝐾 =ℚ(𝛼) where 𝛼 =√−21. alg

Sage

K.<α> = QuadraticField(-21)

Discriminant

By Discriminant of an algebraic integer,

Δ𝐾=−84

Group of units

By ^P1,

O×𝐾={1,−1}.

Class group

Minkowski’s bound is given by

𝑀𝐾=4√21𝜋<6,

so applying Kummer’s factorization theorem

𝑝	𝑥2 +21mod𝑝	⟨𝑝⟩	norms
2	(𝑥 +1)2	𝔭2	2
3	𝑥2	𝔭23	3
5	(𝑥 +2)(𝑥 +3)	𝔭5𝔭′5	5,5

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

𝑡	N𝐾:ℚ⁡(𝛼 +𝑡)
±1	2 ⋅11
±2	52
±3	2 ⋅3 ⋅5

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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