## 305 -Extensions by radicals (2)

March 5, 2009

Last lecture we defined ${{\mathbb F}^{p(x)}}$ where ${{\mathbb F}}$ is a subfield of a field ${{\mathbb K},}$ all the roots of the polynomial ${p(x)}$ are in ${{\mathbb K},}$ and all the coefficients of ${p(x)}$ are in ${{\mathbb F}.}$ Namely, if ${r_1,\dots,r_n}$ are the roots of ${p,}$ then ${{\mathbb F}^{p(x)}={\mathbb F}(r_1,\dots,r_n),}$ the field generated by ${r_1,\dots,r_n}$ over ${{\mathbb F}.}$

The typical examples we will consider are those where ${{\mathbb F}={\mathbb Q},}$ ${{\mathbb K}={\mathbb C},}$ and the coefficients of ${p(x)}$ are rational or in fact, integers.

## 580 -Cardinal arithmetic (8)

March 5, 2009

4. Large cardinals and cardinal arithmetic

In section 3 we saw how the powers of singular cardinals (or, at least, of singulars of uncountable cofinality) satisfy strong restrictions. Here I show that similar restrictions hold at large cardinals. There is much more than one could say about this topic, and the results I present should be seen much more like an invitation than a full story. Also, for lack of time, I won’t motivate the large cardinals we will discuss. (In the ideal world, one should probably say a few words about one’s beliefs in large cardinals, since their existence and even their consistency goes beyond what can be done in the standard system ${{\sf ZFC}.}$ I’ll however take their existence for granted, and proceed from there.)

1. Measurable cardinals

Definition 1 ${\kappa}$ is a measurable cardinal iff ${\kappa>\omega}$ and there is a nonprincipal ${\kappa}$-complete ultrafilter over ${\kappa.}$