305 -Fields (2)

February 13, 2009

We continue from last lecture. Examination of a few particular cases finally allows us to complete Example 6. The answer to whether {\mathbb Z}_n (with the operations and constants defined last time) is a field splits into three parts. The first is straightforward.

Lemma 10. For all positive integers n>1, {\mathbb Z}_n satisfies all the properties of fields except possible the existence of multiplicative inverses. \Box.

This reduces the question of whether {\mathbb Z}_n is a field to the problem of finding multiplicative inverses, which turns out to be related to properties of n.
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580 -Cardinal arithmetic (4)

February 11, 2009

2. Silver’s theorem.

From the results of the previous lectures, we know that any power \kappa^\lambda can be computed from the cofinality and gimel functions (see the Remark at the end of lecture II.2). What we can say about the numbers \gimel(\lambda) varies greatly depending on whether \lambda is regular or not. If \lambda is regular, then \gimel(\lambda)=2^\lambda. As mentioned on lecture II.2, forcing provides us with a great deal of freedom to manipulate the exponential function \kappa\mapsto 2^\kappa, at  least for \kappa regular. In fact, the following holds:

Theorem 1. (Easton). If {\sf GCH} holds, then for any definable function F from the class of infinite cardinals to itself such that:

  1. F(\kappa)\le F(\lambda) whenever \kappa\le\lambda, and
  2. \kappa<{\rm cf}(F(\kappa)) for all \kappa,

there is a class forcing {\mathbb P} that preserves cofinalities and such that in the extension by {\mathbb P} it holds that 2^\kappa=F^V(\kappa) for all regular cardinals \kappa; here, F^V is the function F as computed prior to the forcing extension. \Box

For example, it is consistent that 2^\kappa=\kappa^{++} for all regular cardinals \kappa (as mentioned last lecture, the same result is consistent for all cardinals, as shown by Foreman and Woodin, although their argument is significantly more elaborate that Easton’s). There is almost no limit to the combinations that the theorem allows: We could have 2^\kappa=\kappa^{+16} whenever \kappa=\aleph_\tau is regular and \tau is an even ordinal, and 2^\kappa=\kappa^{+17} whenever \kappa=aleph_\tau for some odd ordinal \tau. Or, if there is a proper class of weakly inaccessible cardinals (regular cardinals \kappa such that \kappa=\aleph_\kappa) then we could have 2^\kappa= the third weakly inaccessible strictly larger than \kappa, for all regular cardinals \kappa, etc.

Morally, Easton’s theorem says that there is nothing else to say about the gimel function on regular cardinals, and all that is left to be explored is the behavior of \gimel(\lambda) for singular \lambda. In this section we begin this exploration. However, it is perhaps sobering to point out that there are several weaknesses in Easton’s result.

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305 -4. Fields.

February 11, 2009

Definition 1. Let {\mathbb F} be a set. We say that the quintuple ({\mathbb F},+,\times,0,1) is a field iff the following conditions hold:

  1. +:{\mathbb F}\times {\mathbb F}\to {\mathbb F}.
  2. \times:{\mathbb F}\times {\mathbb F}\to {\mathbb F.} (We say that {\mathbb F} is closed under addition and multiplication.)
  3. 0,1\in{\mathbb F}.
  4. 0\ne1.
  5. Properties 1–9 of the Theorem from last lecture hold with elements of {\mathbb F} in the place of complex numbers, {}0 in the place of \hat0, and {}1 in the place of \hat1.

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305 -Homework set 2

February 9, 2009

This set is due February 20 at the beginning of lecture. Consult the syllabus for details on the homework policy. I do not think this set is particularly difficult, but it is on the longish side of things, so make sure you leave yourself enough time to work on it.

1. Gauß’ fundamental theorem of algebra states that any equation p(x)=0, where p is a polynomial with complex coefficients, has at least one complex root z. This means that z is a complex number and p(z)=0. Show that p has at most n roots, where n is its degree, and that if we count roots up to multiplicity, then it has exactly n roots. Since the multiplicity of a root z is by definition the largest m such that (x-z)^m is a factor of p(x), you may want to verify that p(z)=0 iff (x-z) is a factor of p.  

2. Let p(x) be a polynomial with real coefficients, and let z be a complex root of p. Show that p(\bar z)=0 as well. Conclude that if the degree n of p is odd and the coefficients of p are real, then p has at least one real root. (You may use the fundamental theorem of algebra, if needed.) Conclude also that if p is of degree four and has real coefficients, then p can be factored as the product of two quadratic polynomials with real coefficients. (Does this follow “directly” from the argument described in lecture?)

3. Solve exercises 54-56 from Chapter 3 of the book.

4. Show directly that if a,b,c are real numbers, then at least one of the solutions of x^3+ax^2+bx+c=0 is a real number. What I mean is that, rather than appealing to problem 2, you want to look at the solutions obtained by Cardano’s method as described in lecture, and argue directly from the formulas so obtained that at least one of the solutions must be real. Be careful, since your argument should not give you that all three roots are real, since this is not true in general.

5. Show directly that a quartic with complex coefficients admits only 4 roots. What I mean is that, rather than appealing to problem 1, you want to look at the solutions obtained by Ferrari’s method as described in lecture, and argue directly that they only produce 4 roots, even though, in principle, they produce 24 (since they involve solving a cubic and then taking a square root to obtain parameters from which four solutions are then found).

305 -3. Complex numbers.

February 9, 2009

Mathematicians first approached complex numbers cautiously. Although it was clear that they were useful in solving certain problems at least formally (for example, they are needed to even make sense of the formulas we found in the previous lectures) what was not clear was that they made sense. Perhaps indiscriminate use of them would lead to contradictions.

Gauß solved this problem by realizing that one can define {\mathbb C} and its operations in terms of {\mathbb R} and its operations. As long as we are willing to accept that {\mathbb R} makes sense, then no contradictions will come up from the use of complex numbers.

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580 -Cardinal arithmetic (3)

February 9, 2009

It is easy to solve negatively the question immediately following Homework problem 5 on lecture II.1. I asked whether if X is Dedekind-finite but {\mathcal P}(X) is Dedekind-infinite, then it followed that there is an infinite Dedekind-finite set Y such that {\mathcal P}(Y)\preceq X.

To exhibit a counterexample, it is enough to know that it is consistent to have an infinite Dedekind finite set X that is the countable union of finite sets (in fact, sets of size 2). Notice that \omega is a surjective image of X, so {\mathcal P}(X) is Dedekind-infinite. Suppose that {\mathcal P}(Y)\preceq X. Then certainly Y\preceq X, so Y is a countable union of finite sets Y_n. If Y is infinite then Y_n\ne\emptyset for infinitely many values of n. But then \omega is also a surjective image of Y, so \omega (and in fact P(\omega)) injects into {\mathcal P}(Y) and therefore into X, contradiction.

At the end of last lecture we showed Theorem 10, a general result that allows us to compute products \kappa^\lambda for infinite cardinals \kappa,\lambda, namely:

Let \kappa and \lambda be infinite cardinals. Let \tau=\sup_{\rho<\kappa}|\rho|^\lambda. Then 

\displaystyle \kappa^\lambda=\left\{\begin{array}{cl} 2^\lambda & \mbox{if }\kappa\le 2^\lambda,\\ \kappa\cdot\tau & \mbox{if }\lambda<{\rm cf}(\kappa),\\ \tau & \begin{array}{l}\mbox{if }{\rm cf}(\kappa)\le\lambda,2^\lambda<\kappa,\mbox{ and }\\ \rho\mapsto|\rho|^\lambda\mbox{ is eventually constant below }\kappa,\end{array}\\ \kappa^{{\rm cf}(\kappa)} & \mbox{otherwise.}\end{array}\right.

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580 -Cardinal arithmetic (2)

February 7, 2009

At the end of last lecture, we showed Theorem 7, König’s lemma, stating that if \kappa_i<\lambda_i for all i\in I, then \sum_{i\in I}\kappa_i<\prod_i\lambda_i. We begin by looking at some  corollaries:

Corollary 8.

  1. If \beta is a limit ordinal and (\kappa_i:i<\beta) is a strictly increasing sequence of nonzero cardinals, then \sum_{\alpha<\beta}\kappa_\alpha<\prod_{\alpha<\beta}\kappa_\alpha.
  2. If (\kappa_i:i\in I) is an I-indexed sequence of nonzero cardinals and \kappa_i<\sum_{j\in I}\kappa_j for all i\in I, then \sum_i\kappa_i<\left(\sum_i\kappa_i\right)^{|I|}.
  3. (Cantor) \kappa<2^\kappa.
  4. For any infinite \kappa, one has \kappa<\kappa^{{\rm cf}(\kappa)}.
  5. {\rm cf}(2^\kappa)>\kappa.

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