ABSTRACT
27.1 It may be a part of human condition that once you have something you look for something bigger! Nevertheless it is a mathematical fact that substructures are harder to study than extensions; indeed very few properties descend from a ring to a subring or from a group to a subgroup, whereas extensions seem to be easier to describe. The same is true for fields and probably this is due to the way that properties of interest have been defined in the first place; for example, it is easy to describe a purely transcendental extension of degree 1 of a field k(they are all isomorphic), but what about the subfields of k such that k is purely transcendental of degree one over them ? It should be pointed out that several open problems in Algebra today are related with “transcendence” e.g. rationality problems in Algebraic Geometry or in the theory of the Brauer group. Algebraic extensions are much better understood ! These extensions have an immediate link to polynomials over the ground field, because an algebraic element is a zero of an irreducible polynomial; then the theory developed in Chapter 22, Chapter 23 and Chapter 25 comes in very handy (see Theorem 27.11, Definition 27.15 etc.). Another trivial advantage of extensions is that there exists algebraically closed extensions for any field, see Theorem 28.94, (while it is easy to see that k need not contain any subfield it is the algebraic closure of!). The Fundamental Theorem of Algebra (27.26, 28.92) yields an example of an algebraically closed field : ℂ, the complex numbers. Since ℂ is also complete this puts the complex numbers in a very central position between Algebra and Analysis, but also in Analytical as well as Algebraic Geometry. At the other extreme ℚ allows many algebraic extensions, therefore Number Theory is so rich and complex. We include at the end of this chapter some examples of algebraic extensions of ℚ mainly 464to highlight the utility of the Degree Formula given in Theorem 27.29.
