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If it is bigger than 1 for some j, the field extension L/K is called ramified at p (or we say that p ramifies in L, or that it is ramified in L). Otherwise, L / K is called unramified at p . If this is the case then by the Chinese remainder theorem the quotient O L / pO L is a product of fields F j .
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In mathematics, the ramification theory of valuations studies the set of extensions of a valuation v of a field K to an extension L of K. It is a generalization of the ramification theory of Dedekind domains. [1] [2] The structure of the set of extensions is known better when L/K is Galois.
The more detailed analysis of ramification in number fields can be carried out using extensions of the p-adic numbers, because it is a local question. In that case a quantitative measure of ramification is defined for Galois extensions , basically by asking how far the Galois group moves field elements with respect to the metric.
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