Modern Approach By Field Theory
In the modern approach, one starts with a field extension L/K (read: L over K), and examines the group of field automorphisms of L/K (these are mappings α: L → L with α(x) = x for all x in K). See the article on Galois groups for further explanation and examples.
The connection between the two approaches is as follows. The coefficients of the polynomial in question should be chosen from the base field K. The top field L should be the field obtained by adjoining the roots of the polynomial in question to the base field. Any permutation of the roots which respects algebraic equations as described above gives rise to an automorphism of L/K, and vice versa.
In the first example above, we were studying the extension Q(√3)/Q, where Q is the field of rational numbers, and Q(√3) is the field obtained from Q by adjoining √3. In the second example, we were studying the extension Q(A,B,C,D)/Q.
There are several advantages to the modern approach over the permutation group approach.
- It permits a far simpler statement of the fundamental theorem of Galois theory.
- The use of base fields other than Q is crucial in many areas of mathematics. For example, in algebraic number theory, one often does Galois theory using number fields, finite fields or local fields as the base field.
- It allows one to more easily study infinite extensions. Again this is important in algebraic number theory, where for example one often discusses the absolute Galois group of Q, defined to be the Galois group of K/Q where K is an algebraic closure of Q.
- It allows for consideration of inseparable extensions. This issue does not arise in the classical framework, since it was always implicitly assumed that arithmetic took place in characteristic zero, but nonzero characteristic arises frequently in number theory and in algebraic geometry.
- It removes the rather artificial reliance on chasing roots of polynomials. That is, different polynomials may yield the same extension fields, and the modern approach recognizes the connection between these polynomials.
Read more about this topic: Galois Theory
Famous quotes containing the words modern, approach, field and/or theory:
“The experience of the gangster as an experience of art is universal to Americans. There is almost nothing we understand better or react to more readily or with quicker intelligence.... In ways that we do not easily or willingly define, the gangster speaks for us, expressing that part of the American psyche which rejects the qualities and the demands of modern life, which rejects “Americanism” itself.”
—Robert Warshow (1917–1955)
“To approach a city ... as if it were [an] ... architectural problem ... is to make the mistake of attempting to substitute art for life.... The results ... are neither life nor art. They are taxidermy.”
—Jane Jacobs (b. 1916)
“The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric which impinges on experience only along the edges. Or, to change the figure, total science is like a field of force whose boundary conditions are experience.”
—Willard Van Orman Quine (b. 1908)
“... liberal intellectuals ... tend to have a classical theory of politics, in which the state has a monopoly of power; hoping that those in positions of authority may prove to be enlightened men, wielding power justly, they are natural, if cautious, allies of the “establishment.””
—Susan Sontag (b. 1933)