Weil Group of A Number Field
For number fields there is no known "natural" construction of the Weil group without using cocycles to construct the extension. The map from the Weil group to the Galois group is surjective, and its kernel is the connected component of the identity of the Weil group, which is quite complicated.
Read more about this topic: Weil Group
Famous quotes containing the words weil, group, number and/or field:
“When a mans life is destroyed or damaged by some wound or privation of soul or body, which is due to other mens actions or negligence, it is not only his sensibility that suffers but also his aspiration toward the good. Therefore there has been sacrilege towards that which is sacred in him.”
—Simone Weil (19091943)
“The conflict between the need to belong to a group and the need to be seen as unique and individual is the dominant struggle of adolescence.”
—Jeanne Elium (20th century)
“The two great points of difference between a democracy and a republic are: first, the delegation of the government, in the latter, to a small number of citizens elected by the rest; secondly, the greater number of citizens and greater sphere of country over which the latter may be extended.”
—James Madison (17511836)
“The head must bow, and the back will have to bend,
Wherever the darkey may go;
A few more days, and the trouble all will end,
In the field where the sugar-canes grow.
A few more days for to tote the weary load,
No matter, t will never be light;
A few more days till we totter on the road:
Then my old Kentucky home, good-night!”
—Stephen Collins Foster (18261884)