Timeline for Is there a number field $K$ such that $K^\times / \mathbb{Q}^\times$ is finitely generated? [duplicate]

6 events

when toggle format	what		by	license	comment
Oct 2, 2022 at 8:31	comment	added	YCor		Maybe there are refinements of this result. E.g., one could ask whether this group always has infinite $\mathbf{Q}$-rank (this time, when $K\neq L$ and $L$ is not algebraic over a finite field). This might follow from the given argument but I'm not competent enough to see this clearly.
Oct 2, 2022 at 8:28	comment	added	YCor		As follows from this answer, for a field extension $K\subset L$, the group $L^/K^$ is finitely generated iff $K=L$ or $L$ is finite (i.e., never except in trivial cases).
Oct 2, 2022 at 7:47	history	closed	GH from MO abx Derek Holt gr.group-theory		Duplicate of If L is a field extension of K, how big is L/K?
Oct 2, 2022 at 6:51	review	Close votes
Oct 2, 2022 at 7:51
Oct 2, 2022 at 6:05	comment	added	user127776		You can assume the extension is Galois. Assuming that the degree of the extension is $n>1$. You still need to use the norm map. You also need to show that infinite number of primes appear as a factor in the image of the norm map that its exponent is not divisible by $n$. For this you just need to show infinite number of primes do not split in $\mathcal{O}_K$.
Oct 2, 2022 at 4:18	history	asked	Bma	CC BY-SA 4.0