# Ideal Class Group of $\mathbb{Q}(\sqrt[3]{3})$

I'm trying to compute the ideal class group of $$\mathbb Q(\sqrt[3]{3})$$, and I would like to know if my calculations are right and if I could improve my arguments.

Let $$K=\mathbb Q (\sqrt[3]{3}),$$ its ring of integers $$\mathcal O_K$$ is $$\mathbb Z[\sqrt[3]{3}]$$, since Disc$$(1,\sqrt[3]{3},\sqrt[3]{3^2})=-3^5 \quad$$ and $$f(x)=x^3-3 \$$ is Eisenstein for $$3$$.

The Minkowski bound is $$\frac{4}{\pi}\cdot\frac{6}{27}\cdot \sqrt{3^5}<5$$. Thus we need to check ideals of norm less than $$5$$. Set $$a=\sqrt[3]{3}$$ to facilitate notation.

The only ideal of norm 1 is $$\mathcal{O}_K$$

Every ideal contains its norm, thus ideals of norm $$2$$ are prime factors of $$(2). \quad$$ $$(2)=(a-1)(a^2+a+1) \$$, since $$2=3-1=a^3-1=(a-1)(a^2+a+1). N(a-1)=2, N(a^2+a+1)=4$$.

$$(3)=(a)^3$$ and $$(a)$$ is principal.

Since $$(2)=(a-1)(a^2+a+1),$$ ideals of norm 4 are: $$(a-1)^2, \ (a^2+a+1).$$

All of the ideals above are principal, thus the ideal class group is trivial and $$\mathcal O_K$$ is a PID.

Your computations are correct and they seem quite optimal. Here is a verification in SageMath:

sage: x = polygen(QQ)
sage: f = x^3 - 3
sage: f.is_irreducible()
True
sage: K.<a> = NumberField(f); K
Number Field in a with defining polynomial x^3 - 3
sage: OK = K.ring_of_integers(); OK
Maximal Order in Number Field in a with defining polynomial x^3 - 3
sage: OK.basis()
[1, a, a^2]
sage: OK.discriminant().factor()
-1 * 3^5
sage: M = K.minkowski_bound(); M
8*sqrt(3)/pi
sage: bool(M == 4/pi * 6/27 * sqrt(3^5))
True
sage: bool(M < 5)
True
sage: OK.ideal(2).factor()
(Fractional ideal (a^2 + a + 1)) * (Fractional ideal (a - 1))
sage: OK.ideal(3).factor()
(Fractional ideal (a))^3
sage: K.class_number()
1
sage: OK.class_group()
Class group of order 1 of Number Field in a with defining polynomial x^3 - 3


Of course you meant to say $$\mathcal{O}_K$$ is a PID. ($$K$$ is also always a PID.)

• Thanks a lot for your help! I should definetely learn how to use sage, and thanks for spotting the typo! Feb 14, 2021 at 20:28