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0answers
79 views

Is there a Kummer theory for cyclic covers in higher dimension?

Let $d \geq 2$ be an integer and $K$ a field of characteristic zero containing the roots of unity of order $d$. Let $X$ be a smooth curve over $K$, not necessarily projective. Let $\overline{x}$ be a ...
6
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1answer
253 views

Degree of Kummer extensions of number fields

Let $K$ be a number field and $a\in K^*$ of infinite order in $K^*$. How do I show that $$[K(\sqrt[n]{a},\zeta_n):K]\geq C\cdot n\cdot\varphi(n)$$ holds for all positive integers $n$, with a positive ...
3
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1answer
412 views

Kummer theory isomorphism and Kummer extensions

Let $p$ be a prime number and $K$ a finite extension of $\mathbb{Q}_p$. Put $K_\infty = K(\mu_{p^\infty})$, the field extension obtained by adjoining all $p$-power roots of unity to $K$. I want to ...
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0answers
200 views

a reference for Kummer theory, with proofs ?

What is a standard reference for Kummer theory of semi-Abelian varieties ? I need a complete exposition with detailed proofs. Also in prime characteristic, although I am not sure what the statement ...