Normal Bases in Galois Theory
In Galois theory it is shown that for a field L, and a finite group G of automorphisms of L, the fixed field K of G has = |G|. In fact we can say more: L viewed as a K-module is the regular representation. This is the content of the normal basis theorem, a normal basis being an element x of L such that the g(x) for g in G are a vector space basis for L over K. Such x exist, and each one gives a K-isomorphism from L to K. From the point of view of algebraic number theory it is of interest to study normal integral bases, where we try to replace L and K by the rings of algebraic integers they contain. One can see already in the case of the Gaussian integers that such bases may not exist: a + bi and a − bi can never form a Z-module basis of Z because 1 cannot be an integer combination. The reasons are studied in depth in Galois module theory.
Read more about this topic: Regular Representation
Famous quotes containing the words normal, bases and/or theory:
“Everyone in the full enjoyment of all the blessings of his life, in his normal condition, feels some individual responsibility for the poverty of others. When the sympathies are not blunted by any false philosophy, one feels reproached by one’s own abundance.”
—Elizabeth Cady Stanton (1815–1902)
“The information links are like nerves that pervade and help to animate the human organism. The sensors and monitors are analogous to the human senses that put us in touch with the world. Data bases correspond to memory; the information processors perform the function of human reasoning and comprehension. Once the postmodern infrastructure is reasonably integrated, it will greatly exceed human intelligence in reach, acuity, capacity, and precision.”
—Albert Borgman, U.S. educator, author. Crossing the Postmodern Divide, ch. 4, University of Chicago Press (1992)
“every subjective phenomenon is essentially connected with a single point of view, and it seems inevitable that an objective, physical theory will abandon that point of view.”
—Thomas Nagel (b. 1938)