305 -Extensions by radicals (2)

March 5, 2009

2. Extensions by radicals

 

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.

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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.}

 

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