Prove if F(\sqrt[n]{a}) is unramified or totally ramified in certain

hadaasyj 2022-04-28 Answered

Prove if F(an) is unramified or totally ramified in certain conditions

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Answers (1)

Tatairfzk
Answered 2022-04-29 Author has 12 answers
Step 1
For (1) xna is separable in the residue field OF(πF) so F(a1n)F is automatically unramified.
Note that Hensel lemma gives that ζq1F(a1n) where q is the cardinality of the residue field, and xna is separable in the residue field so Hensel lemma again gives that a1nF(ζq1) and F(a1n)=F(ζq1).
For (2) take nl+mv(a)=1, let b=amπFnl, v(b)=1, F(a1n)=F(b1n) and xnb is Eisenstein over OF so F(b1n)F has degree n and v(b1n)=1n, it is totally ramified.
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