## 305 -6. Rings, ideals, homomorphisms

March 13, 2009

It will be important to understand the subfields of a given field; this is a key step in figuring out whether a field ${{\mathbb Q}^{p(x)}}$ is an extension by radicals or not. We need some “machinery” before we can develop this understanding.

Recall:

Definition 1 A ring is a set ${R}$ together with two binary operations ${+,\times}$ on ${R}$ such that:

1. ${+}$ is commutative.
2. There is an additive identity ${0.}$
3. Any ${a}$ has an additive inverse ${-a.}$
4. ${+}$ is associative.
5. ${\times}$ is associative.
6. ${\times}$ distributes over ${+,}$ both on the right and on the left.

## 580 -Cardinal arithmetic (12)

March 13, 2009

5. PCF theory

To close the topic of cardinal arithmetic, this lecture is a summary introduction to Saharon Shelah’s pcf theory. Rather, it is just motivation to go and study other sources; there are many excellent references available, and I list some below. Here I just want to give you the barest of ideas of what the theory is about and what kinds of results one can achieve with it. All the results mentioned are due to Shelah unless otherwise noted. All the notions mentioned are due to Shelah as far as I know.

Some references:

• Maxim Burke, Menachem Magidor, Shelah’s pcf theory and its applications, Annals of pure and applied logic, 50, (1990), 207–254.
• Thomas Jech, Singular cardinal problem: Shelah’s theorem on ${2^{\aleph_\omega}}$, Bulletin of the London Mathematical Society, 24, (1992), 127–139.
• Saharon Shelah, Cardinal arithmetic for skeptics, Bulletin of the American Mathematical Society, 26 (2), (1992), 197–210.
• Saharon Shelah, Cardinal arithmetic, Oxford University Press, (1994).
• Menachem Kojman, The ABC of pcf, unpublished notes, available (as of this writting) at his webpage.
• Uri Abraham, Menachem Magidor, Cardinal arithmetic, in Handbook of set theory, Matthew Foreman, Akihiro Kanamori, eds., forthcoming.
• Todd Eisworth, Successors of singular cardinals, in Handbook of set theory, Matthew Foreman, Akihiro Kanamori, eds., forthcoming.