Solve exercises 7, 35, 40, 43, 44, 45 from Chapter 6 of the book.
Last lecture we characterized subfields and used the characterization to provide many new examples of fields. Now we start to explore systematically which subfields of the complex numbers are suitable to study the question of which polynomial equations can be solved. Read the rest of this entry »
A. The Galvin-Hajnal rank and an improvement of Theorem 3.1
Last lecture, I covered the first theorem of the Galvin-Hajnal paper and several corollaries. Recall that the result, Theorem 3.1, states that if and are uncountable regular cardinals, and is -inaccessible, then for any sequence of cardinals such that for all
In particular (see, for example, Corollary 3.7), if and is strong limit, then
The argument relied in the notion of an almost disjoint transversal. Assume that is regular and uncountable, and recall that if is a sequence of sets, then is an a.d.t. for Here, is an a.d.t. for iff and whenever then is bounded.
With as above, Theorem 3.1 was proved by showing that there is an a.d.t. for of size and then proving that, provided that for all then
In fact, the argument showed a bit more. Recall that if then Then, for any ,
The proof of this result was inductive, taking advantage of the well-foundedness of the partial order defined on by iff is bounded in That is well-founded allows us to define a rank for each and we can argue by considering a counterexample of least possible rank to the statement from the previous paragraph.
In fact, more precise results are possible. Galvin and Hajnal observed that replacing the ideal of bounded sets with the nonstationary ideal (or, really, any normal ideal), results in a quantitative improvement of Theorem 3.1. Read the rest of this entry »