Imaginary quadratic field

ℚ(√−14)

Consider the monogenic imaginary quadratic field 𝐾 =ℚ(𝛼) where 𝛼 =√−14. alg

Sage

K.<α> = QuadraticField(-14)

Discriminant

By Discriminant of an algebraic integer,

Δ𝐾=−56.

Group of units

By ^P1,

O×𝐾={1,−1}.

Class group

Minkowski’s bound is given by

𝑀𝐾=4√14𝜋<5,

so applying Kummer’s factorization theorem

𝑝	𝑥2 +14mod𝑝	⟨𝑝⟩	norms
2	𝑥2	𝔭22	2
3	(𝑥 +1)(𝑥 −1)	𝔭3𝔭′3	3,3

Clearly no algebraic integers can have these norms, so we can be satisfied that these are not principal. Since 𝔭−13 =𝔭′3, the ideal class group is generated by {[𝔭2],[𝔭3]}. Some algebraic integers of small field norm are

𝑡	N𝐾:ℚ⁡(𝛼 +𝑡)
±1	3 ⋅5
±2	2 ⋅32
±3	23
from which we derive the relation

𝔭2𝔭23=⟨2,𝛼⟩⟨3,𝛼+1⟩2=⟨𝜗−2⟩∼⟨1⟩

whence 𝔭23 ∼𝔭2, so 𝔭43 ∼⟨1⟩. Therefore

Cl⁡𝐾=⟨[𝔭3]⟩≅C4.

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