2 votes

Possible failure of $ef=n$ for a degree $n$ extension of a field complete to a nondiscrete nonarchimedean valuation?

Let $T$ be the maximal tamely ramified extension of $\mathbb{Q}_p$ with $p>2$, and $K$ be the completion of $T_p=T(\mu_{p^{\infty}})$. It’s clear that $|K^{\times}|=p^{\mathbb{Q}}$: in particular, ...
  • 30.8k
2 votes

Residue field of DVR

Talking about DVRs immediately is the wrong way to solve this problem. The right way is to use the fact that you're dealing with a scheme of finite type over $k$: Suppose $C$ is our curve and let $U\...
  • 56.6k
1 vote

Find the possible values for $[\mathbb{Q}(\alpha,\beta):\mathbb{Q}]$ and provide an example of $(\alpha,\beta)$ for each possible value

If $\left[\mathbb Q[\alpha,\beta]:\mathbb Q[\alpha]\right]=3,$ then $\beta$ must have a minimal polynomial some cubic monic polynomial $p(x)$ in $\mathbb Q[\alpha][x].$ But $x$ has a minimal ...

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