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Let $(K,\nu)$ be a local field whit characteristic $0$ and residue field $k=\mathbb{F}_q$, $q=p^f$ whit $p$ prime. I have to prove that there is a finite number of isomorphism classes of extension $K’$ of $K$ such that $[K’:K]=n$. I have no idea how to prove it, but I have two advice:

  1. $O_K$ is a compact group
  2. Corollary of Krasner’s lemma: Let $f(x)=xⁿ+a_1x^{n-1}+…+a_n,g(x)=xⁿ+b_1x^{n-1}+…+b_n \in O_K[x]$ be polynomial whit $f(x)$ irreducible polynomial. If exist $M>0$ Such that $\nu(a_i-b_i)>M>n \cdot max\{\nu ‘(\alpha_i-\alpha_j|1 \leq i \neq j \leq n\}$,whit $\alpha_1,…,\alpha_n$ zeros of $f(x)$ in a Galois extension $K’$ of $K$ and $\nu’$ is the valuation on $K’$, then $g(x)$ is irreducible polynomial and $K[x]/(f(x)) \cong K[x]/(g(x))$

Anyone have any ideas?

Mario
  • 742
  • How much do you know about local fields? It seems tough to do this directly, but an approach works where you break an extension into an unramified and totally ramified extension and then just count that there are finitely many totally ramified extensions (since there is always just one unramified extension of a given degree of any local field), the latter you relate the totally ramified extensions to Eisenstein polynomials, see here https://math.stackexchange.com/questions/1118068/finitely-many-extensions-of-fixed-degree-of-a-local-field – TY Mathers Jul 20 '22 at 23:03

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