Fields With Finite Absolute Galois Groups
Let K be a field and let G = Gal(K) be its absolute Galois group. If K is algebraically closed, then G = 1. If K = R is the real numbers, then
Here C is the field of complex numbers and Z is the ring of integer numbers. A theorem of Artin and Schreier asserts that (essentially) these are all the possibilities for finite absolute Galois groups.
Artin–Schreier theorem. Let K be a field whose absolute Galois group G is finite. Then either K is separably closed and G is trivial or K is real closed and G = Z/2Z.
Read more about this topic: Field Arithmetic
Famous quotes containing the words fields, finite, absolute and/or groups:
“Come up from the fields father, here’s a letter from our Pete,
And come to the front door mother, here’s a letter from thy dear
son.”
—Walt Whitman (1819–1892)
“For it is only the finite that has wrought and suffered; the infinite lies stretched in smiling repose.”
—Ralph Waldo Emerson (1803–1882)
“The absolute things, the last things, the overlapping things, are the truly philosophic concerns; all superior minds feel seriously about them, and the mind with the shortest views is simply the mind of the more shallow man.”
—William James (1842–1910)
“Under weak government, in a wide, thinly populated country, in the struggle against the raw natural environment and with the free play of economic forces, unified social groups become the transmitters of culture.”
—Johan Huizinga (1872–1945)