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Let us consider the rational number field $ \mathbb{Q}$ and its finite extension $ K$.

I can't understand the concept of large ramification index or enough Ramification or sufficient Ramification.

Let $A$ be Dedekind domain, $K$ be its fraction field and let $L$ be finite extension of $K$ of degree $n$. Let $B$ be the integral closure of $A$.

Let $P$ be a nonzero prime ideal of $A$. The lofting (also called extension) of $P$ to $B$ is the ideal $PB$. Now the ideal $PB$ may not be prime ideal of $B$ but since it is a Dedekind domain , we can express it as a product of prime ideals in $B$ i.e., $$ PB=\prod_{i=1}^{g} P_i^{e_i} $$ where $P_i$ are distinct prime ideals of $B$ and $e_i$ are positive integers.

This Phenomenon is called Ramification.

But my question is what is Enough Ramification or large ramification (index)?

Please explain this fact with one example.

Thanks

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    $\begingroup$ Hi! Welcome on Math.SE! In order to help people in answering your question, can you add a definition of ramification to your question? $\endgroup$ – Filippo De Bortoli Sep 6 '18 at 6:17
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    $\begingroup$ That is NOT at all what ramification means! It most definitely has a particular definition, e.g., referring to a prime number factoring in the ring of integers of a number field with a repeated prime ideal factor. $\endgroup$ – KCd Sep 6 '18 at 7:09
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    $\begingroup$ Your edit sets up useful notation, but still does not say what it means for $P$ to ramify in $L$. What you wrote sounds like the phenomenon of factoring is ramification, but that is not correct, e.g., in $\mathbf Z[i]$ we have $(5) = (1+2i)(1-2i)$ so the prime $5$ in $\mathbf Z$ is not prime in $\mathbf Z[i]$, but it us not ramified either. Any algebraic number theory book will correctly define ramification. It feels like you don't know examples when you describe the definition incorrectly. $\endgroup$ – KCd Sep 6 '18 at 7:20
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    $\begingroup$ By the way, your username is misspelled: the math term is Non-Archimedean. $\endgroup$ – KCd Sep 6 '18 at 7:21
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    $\begingroup$ the positive integer $e_i$ is the ramification index of $P_i$. If that number is large we can also say there is a lot of ramification. $\endgroup$ – mercio Sep 6 '18 at 11:49

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