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?


1 Answer 1


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.

  • $\begingroup$ And I guess that if you don’t choose $\pi=p$ in this case, you don’t even get the cyclotomic tower. $\endgroup$
    – Lubin
    Apr 6 at 21:47

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