The **Weil group** of a class formation with fundamental classes *u*_{E/F} ∈ *H*2(*E*/*F*, *A**F*) is a kind of modified Galois group, used in various formulations of class field theory, and in particular in the Langlands program.

If *E*/*F* is a normal layer, then the (relative) Weil group *W _{E}*

_{/F}of

*E*/

*F*is the extension

- 1 →
*A**F*→*W*_{E}_{/F}→ Gal(*E*/*F*) → 1

corresponding (using the interpretation of elements in the second group cohomology as central extensions) to the fundamental class *u*_{E/F} in *H*2(Gal(*E*/*F*), *A**F*). The Weil group of the whole formation is defined to be the inverse limit of the Weil groups of all the layers *G*/*F*, for *F* an open subgroup of *G*.

The reciprocity map of the class formation (*G*, *A*) induces an isomorphism from *AG* to the abelianization of the Weil group.

Read more about Weil Group: Weil Group of An Archimedean Local Field, Weil Group of A Finite Field, Weil Group of A Local Field, Weil Group of A Function Field, Weil Group of A Number Field, Weil–Deligne Group, Langlands Group, See Also

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