if $p\mathcal{O}_K$ splits completely in Galois extension, are all primes lying over $p$ generated by one class in $CL(K)$?

Question

L.S.,

Studying for my exam on algebraic number theory, I was thinking of writing down tricks for computing the class group of a number field fast. I thought of the following one, but I don't know if it is true. Could one maybe help me show whether it is or isn't?

Trick:

Let $K:\mathbb{Q}$ be a Galois extension of the degree $n$. Let $(p)$ split completely in $\mathcal{O}_K$ as $(p) = \prod_{i = 1}^n \mathfrak{p}_i$, with all $\mathfrak{p}_i$ of norm $p$. Now all $[\mathfrak{p}_i] \in CL(K)$ are generated by one $[\mathfrak{p}_j]$ for some $1 \leq j \leq n$.

The reason I suspect this might somehow be true, is because I already know that the Galois group acts transitively on a set of prime ideals lying over some $(p)$, and also because it happened in most class groups I computed so far.

Many thanks!

Answer 1

After some playing around with Sage, here is a counterexample.

Let $K=\mathbb Q(\sqrt[3]{11}, \zeta)$ where $\zeta$ is a cubed root of unity. Then $K$ is Galois with class group $C_2\times C_2$.

The prime $19$ splits completely in $K$. Two of the primes of $K$ lying above $19$ are $$(19, \zeta-\sqrt[3]{11}-2)\qquad\text{ and }\qquad(19, \zeta-\sqrt[3]{11}+9)$$

Neither of these primes are principal. Hence, if their ideal classes are to lie in a cyclic subgroup of $C_2\times C_2$ (which must be of order $2$), then both ideals must be in the same ideal class, and hence their product should be principal. However, their product is not principal.

Sage code:

K.<a> = NumberField(x^3-11)
L.<b> = K.extension(x^2+x+1)
C = L.class_group(); C
OL=L.ring_of_integers()

split =[C(P) for P,e in (19*OL).factor()]
split

split[0].is_principal()
split[1].is_principal()
test = split[0]*split[1]
test.is_principal()

and output:

Class group of order 4 with structure C2 x C2 of Number Field in b with defining polynomial x^2 + x + 1 over its base field
[Fractional ideal class (19, b - a - 2), Fractional ideal class (19, b - a + 9), Fractional ideal class (19, b - a - 9), Fractional ideal class (19, b - a - 6), Fractional ideal class (19, b - a + 5), Fractional ideal class (19, b - a + 6)]
False
False
False