# Field extension corresponding to a quotient of units of local fields

Let $$K$$ be a finite extension of $$\mathbb{Q}_p$$. Let $$\mathcal{O}$$ be its ring of integers and $$\mathfrak{m}$$ the maximal ideal. Pick a uniformiser $$\pi$$. The construction using theory of Lubin--Tate formal groups yields an abelian Galois extension $$K_\pi$$ of $$K$$ such that $$\mathrm{Gal}(K_\pi/K)\simeq\mathcal{O}^\times$$. Note that it is well-known that $$\mathcal{O}^\times\simeq\mu_{q-1}\times\mathbb{Z}/p^a\times\mathbb{Z}_p^d$$ where $$d=[K:\mathbb{Q}_p]$$ and $$q$$ the number of elements of the residue field of $$\mathcal{O}$$ and $$a$$ some integer.

There should be Galois sub-extensions of $$K$$ in $$K_\pi$$ corresponding to each copy of $$\mathbb{Z}_p$$. Is it known what they are?

The isomorphism you write down involving the $$\mathbb{Z}_p$$ is not canonical, so one cannot describe them without fixing additional data. For $$\mathbb{Q}_p$$ you have $$a=1$$ and this tower is the totally ramified (p-)cyclotomic tower.
• And I guess that if you don’t choose $\pi=p$ in this case, you don’t even get the cyclotomic tower. Apr 6 at 21:47