Field Arithmetic - Fields With Finite Absolute Galois Groups

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:

    This scene was supposed to be in a saloon, but the censor cut it out. It’ll play just as well.
    Otis Criblecoblis, U.S. screenwriter. W.C. Fields (W.C. Fields)

    Put shortly, these are the two views, then. One, that man is intrinsically good, spoilt by circumstance; and the other that he is intrinsically limited, but disciplined by order and tradition to something fairly decent. To the one party man’s nature is like a well, to the other like a bucket. The view which regards him like a well, a reservoir full of possibilities, I call the romantic; the one which regards him as a very finite and fixed creature, I call the classical.
    Thomas Ernest Hulme (1883–1917)

    All beauties contain, like all possible phenomena, something eternal and something transitory,—something absolute and something particular. Absolute and eternal beauty does not exist, or rather it is only an abstraction skimmed from the common surface of different sorts of beauty. The particular element of each beauty comes from the emotions, and as we each have our own particular emotions, so we have our beauty.
    Charles Baudelaire (1821–1867)

    In properly organized groups no faith is required; what is required is simply a little trust and even that only for a little while, for the sooner a man begins to verify all he hears the better it is for him.
    George Gurdjieff (c. 1877–1949)