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When / is unramified, by (iv) (or (iii)), G can be identified with (/), which is finite cyclic. The above implies that there is an equivalence of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K.
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.
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 . The extension L / K is ramified in exactly those primes that divide the relative discriminant , hence the extension is unramified in all but finitely many prime ideals.
In algebraic geometry, an unramified morphism is a morphism: of schemes such that (a) it is locally of finite presentation and (b) for each and = (), we have that The residue field k ( x ) {\displaystyle k(x)} is a separable algebraic extension of k ( y ) {\displaystyle k(y)} .
An extension is unramified if, and only if, the discriminant is the unit ideal. [25] The Minkowski bound above shows that there are no non-trivial unramified extensions of Q. Fields larger than Q may have unramified extensions: for example, for any field with class number greater than one, its Hilbert class field is a non-trivial unramified ...
In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K.Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime ideals in K.
The unramified part of any abelian extension is easily constructed, Lubin–Tate finds its value in producing the ramified part. This works by defining a family of modules (indexed by the natural numbers) over the ring of integers consisting of what can be considered as roots of the power series repeatedly composed with itself.
The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. [a] A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension. [2]