580 -III. Partition calculus

March 21, 2009

1. Introduction

Partition calculus is the area of set theory that deals with Ramsey theory; it is devoted to Ramsey’s theorem and its infinite and infinitary generalizations. This means both strengthenings of Ramsey’s theorem for sets of natural numbers (like the Carlson-Simpson or the Galvin-Prikry theorems characterizing the completely Ramsey sets in terms of the Baire property) and for larger cardinalities (like the {\mbox{Erd\H os}}-Rado theorem), as well as variations in which the homogeneous sets are required to possess additional structure (like the Baumgartner-Hajnal theorem).

Ramsey theory is a vast area and by necessity we won’t be able to cover (even summarily) all of it. There are many excellent references, depending on your particular interests. Here are but a few:

  • Paul {\mbox{Erd\H os},} András Hajnal, Attila Máté, Richard Rado, Combinatorial set theory: partition relations for cardinals, North-Holland, (1984).
  • Ronald Graham, Bruce Rothschild, Joel Spencer, Ramsey theory, John Wiley & Sons, second edn., (1990).
  • Neil Hindman, Dona Strauss, Algebra in the Stone-{\mbox{\bf \v Cech}} compactification, De Gruyter, (1998).
  • Stevo {\mbox{Todor\v cevi\'c},} High-dimensional Ramsey theory and Banach space geometry, in Ramsey methods in Analysis, Spiros Argyros, Stevo {\mbox{Todor\v cevi\'c},} Birkhäuser (2005), 121–257.
  • András Hajnal, Jean Larson, Partition relations, in Handbook of set theory, Matthew Foreman, Akihiro Kanamori, eds., to appear.

I taught a course on Ramsey theory at Caltech a couple of years ago, and expect to post notes from it at some point. Here we will concentrate on infinitary combinatorics, but I will briefly mention a few finitary results.

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305 -Homework set 6

March 21, 2009

This set is due April 3 at the beginning of lecture. Details of the homework policy can be found on the syllabus and here.

1. Find {\mathbb Q}^{p(x)} where p(x)=x^3-2, and determine all its subfields. Make sure you justify your answer. For example, if you state that two subfields {\mathbb F}_1 and {\mathbb F}_2 are different, you need to prove that this is indeed the case. 

2. Do the same for p(x)=x^4+x^3+x^2+x+1. 

[Updated, April 2: I guess the hint I gave for problem 2 makes no sense, sorry about that. Rather, you may want to begin by looking at how x^5-1 factors. Then, to compute \cos(72^\circ), it may be helpful to look at a triangle with angles \measuredangle 72^\circ, \measuredangle 72^\circ, and \measuredangle 36^\circ.]


305 -Rings, ideals, homomorphisms (3)

March 21, 2009

In order to understand the construction of the quotient ring from last lecture, it is convenient to examine some examples in details. We are interested in ideals {I} of {{\mathbb F}[x],} where {{\mathbb F}} is a field. We write {{\mathbb F}[x]/I} for the quotient ring, i.e., the set of equivalence classes {[a]_\sim} of polynomials {a} in {F[x]} under the equivalence relation {a\sim b} iff {a-b\in I.}

  • If {I=\{0\},} then for any {a,} the equivalence class {[a]_\sim} is just the singleton {\{a\}} and the homomorphism map {h:{\mathbb F}[x]\rightarrow{\mathbb F}[x]/I} given by {h(a)=[a]_\sim} is an isomorphism.

To understand general ideals better the following notions are useful; I restrict to commutative rings with identity although they make sense in other contexts as well:

Definition 1 Let {R} be a commutative ring with identity. An ideal {I} is principal iff it is the ideal generated by an element {a} of {R,} i.e., it is the set {(a)} of all products {ab} for {b\in R.}

For example, {\{0\}=(0)} is principal. In {{\mathbb Z}} every subring is an ideal and is principal, since all subrings of {{\mathbb Z}} are of the form {n{\mathbb Z}=(n)} for some integer {n.}

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