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2
votes
1answer
135 views

Ramification of prime ideal in Kummer extension

Let $\mu \in \mathbb{Q}(\zeta_n)$ lie above the rational prime $p$, and let the prime ideal $\mathscr{P}\subset \mathbb{Z}[\zeta_n]$ have ramification index $a$ over $\mu$. Why is it then true that ...
3
votes
1answer
109 views

Bibliography suggestion for Kummer theory

I already posted a question about a sum involving the degree of a Kummer extension. Now I'm interested in a more specific fact about Kummer extensions. From Hooley's paper "On Artin's conjecture", we ...
2
votes
1answer
139 views

Doubt concerning a sum involving Kummer extension degrees

I'd like to estimate the following sum $$ \sum_{n\leq x}\frac1{k_n}\;,\qquad x\rightarrow \infty\;, $$ where $k_n=[\mathbb{Q}(\zeta_n,a^{1/n}):\mathbb{Q}]$ is the degree of a Kummer extension for a ...
1
vote
0answers
93 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 ...
7
votes
1answer
344 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
votes
1answer
538 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 ...
2
votes
0answers
242 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 ...