305 -Fields (5)

February 27, 2009

At the end of last lecture we stated a theorem giving an easy characterization of subfields of a given field {mathbb F}. We begin by proving this result.

Theorem 18. Suppose {mathbb F} is a field and Ssubseteq{mathbb F}. If S satisfies the following 5 conditions, then S s a subfield of {mathbb F}:

  1. S is closed under addition.
  2. S is closed under multiplication.
  3. -ain S whenever ain S.
  4. a^{-1}in S whenever ain S and ane0.
  5. S has at least two elements.

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305 -Homework set 4

February 22, 2009

I am not happy with the solutions I received for problems 4 and 5 of Homework set 2 so, for this new set, due March 2 at the beginning of lecture, you must redo these two problems correctly. As usual, the homework policy is detailed in the syllabus. However, there are a few points I want to emphasize:

  • Although I have so far allowed collaboration, each student should write their own solutions. If a group of students collaborate in a problem, they should indicate so at the beginning of their solutions. 

Let me explain this a bit. I do not simply mean that each of you has to write or type your own set of solutions. Of course I expect that, but I expect more than that. When you write your solutions, you should do this on your own. I do not want to see exactly the same mistakes in different people, exactly the same notation, exactly the same equations. If I see it ever again, even if it is not intentional, I will not allow collaboration any longer. Collaboration means that you work together and help one another and give suggestions to one another. Once you have come up with a solution, then collaboration stops and you should write your own version of what was found. 

  • Also, if references are consulted, they should be listed. This means you should mention the books you look at that gave you ideas. This means you should give the name of the book, the author, the edition, the theorem you are using or quoting or being inspired by. Similarly, if you find ideas online, mention the webpage where you found them. 

Now, and this is very important: Your solutions are your own. It may be that by accident in some book you run into the solution of a problem I assigned. This is fine, as long as it is not done intentionally, and I trust your honesty in this regard. Remember that you are bound by an honor code. It is not acceptable to copy the solution the book gives, not even if you give a complete reference that makes it clear that the solution is not your own. If you find a useful idea in a book, make sure you understand it before you use it. Just copying it down will not be acceptable, even if you change notation or the order in which the idea is presented. If you find a solution in a book, and you do not understand it, you will be better off not attempting to use it.  

If I see that the two points above (or any of the details of the homework policy) are not followed, I may change the grading policy and increase the number of quizzes we will have and the percentage they contribute to your total grade. 

Please make sure your solutions are reasonably self contained. If you use a result we have not shown in class you need to provide a proof. Please make sure you turn in your homework on time. This means at the beginning of lecture, not at the end of lecture or in the middle of lecture.

305 -Fields (4)

February 20, 2009

Suppose that {mathbb F} is a field and that Ssubset{mathbb F}. It may be that S is also a field, using the same operations of {mathbb F}. For example, if {mathbb F}={mathbb R}, then we could have S={mathbb Q}.

Definition 15. If {mathbb F} is a field and Ssubset{mathbb F}, we say that S is a subfield of {mathbb F} if S is a field with the operations of {mathbb F}.

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305 -Homework set 3

February 18, 2009

This set is due February 25 at the beginning of lecture. Consult the syllabus for details on the homework policy.

1. Show directly that there is no field of 6 elements. (“Directly” means, among other things, that you cannot use the facts mentioned without proof at the end of lecture 4.3.)

2. Construct a field of size 8. Once you are done, verify that all its elements satisfy the equation x^8-x=0.

3. Solve exercises 36–38 from Chapter 4 of the book.

4. Is the set {a+broot 4of 2 : a,bin{mathbb Q}} a field with the usual +,times,0,1?

305 -Fields (3)

February 18, 2009

At the end of last lecture we arrived at the question of whether every finite field is a {mathbb Z}_p for some prime p.

In this lecture we show that this is not the case, by exhibiting a field of 4 elements. We also find some general properties of finite fields. Finite fields have many interesting applications (in cryptography, for example), but we will not deal much with them as our focus through the course is on number fields, that we will begin discussing next lecture.

We begin by proving the following result:

Lemma 13. Suppose that {mathbb F} is a finite field. Then there is some natural number n>0 such that the sum of n ones vanishes, 1+dots+1=0. The least such n is a prime that divides the size of the field

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580 -Cardinal arithmetic (6)

February 17, 2009

3. The Galvin-Hajnal theorems.

In this section I want to present two theorems of Galvin and Hajnal that greatly generalize Silver’s theorem. I focus on a “pointwise” (or everywhere) result, that gives us information beyond the pointwise theorems from last lecture, like Corollary 23. Then I state a result where the hypotheses, as in Silver’s theorem, are required to hold stationarily rather than everywhere. From this result, the full version of Silver’s result can be recovered.

Both results appear in the paper Fred Galvin, András Hajnal, Inequalities for Cardinal Powers, The Annals of Mathematics, Second Series, 101 (3), (May, 1975), 491–498, available from JSTOR, that I will follow closely. For the notion of kappa-inaccessibility, see Definition II.2.20 from last lecture.  

Theorem 1. Let kappa,lambda be uncountable regular cardinals, and suppose that lambda is kappa-inaccessible. Let (kappa_alpha:alpha<kappa) be a sequence of cardinals such that prod_{alpha<beta}kappa_alpha<aleph_lambda for all beta<kappa. Then also prod_{alpha<kappa}kappa_alpha<aleph_lambda.

The second theorem will be stated next lecture. Theorem 1 is a rather general result; here are some corollaries that illustrate its reach:  

Corollary 2. Suppose that kappa,lambda are uncountable regular cardinals, and that lambda is kappa-inaccessible. Let tau be a cardinal, and suppose that tau^sigma<aleph_lambda for all cardinals sigma<kappa. Then also tau^kappa<aleph_lambda.

Proof. Apply Theorem 1 with kappa_alpha=tau for all alpha<kappa. {sf QED}

Corollary 3. Suppose that kappa,lambda are uncountable regular cardinals, and that lambda is kappa-inaccessible. Let tau be a cardinal of cofinality kappa, and suppose that 2^sigma<aleph_lambda for all cardinals sigma<tau. Then also 2^tau<aleph_lambda.

Proof. Let (tau_alpha:alpha<kappa) be a sequence of cardinals smaller than tau such that tau=sum_alphatau_alpha, and set kappa_alpha=2^{tau_alpha} for all alpha<kappa. Then prod_{alpha<beta}kappa_alpha=2^{sum_{alpha<beta}tau_alpha}<aleph_lambda for all beta<kappa, by assumption. By Theorem 1, prod_{alpha<kappa}kappa_alpha=2^{sum_alphatau_alpha}=2^tau<aleph_lambda as well. {sf QED}    

Corollary 4. Let kappa,rho,tau be cardinals, with rhoge2 and kappa regular and uncountable. Suppose that tau^sigma<aleph_{(rho^kappa)^+} for all cardinals sigma<kappa. Then also tau^kappa<aleph_{(rho^kappa)^+}.

Proof. This follows directly from Corollary 2, since lambda=(rho^kappa)^+ is regular and kappa-inaccessible. {sf QED}

Corollary 5. Let rho,tau be cardinals, with rhoge2 and tau of uncountable cofinality kappa. Suppose that 2^sigma<aleph_{(rho^kappa)^+} for all cardinals sigma<tau. Then also 2^tau<aleph_{(rho^kappa)^+}.

Proof. This follows directly from Corollary 3 with lambda=(rho^kappa)^+. {sf QED}

Corollary 6. Let xi be an ordinal of uncountable cofinality, and suppose that 2^{aleph_alpha}<aleph_{(|xi|^{{rm cf}(xi)})^+} for all alpha<xi. Then also 2^{aleph_xi}<aleph_{(|xi|^{{rm cf}(xi)})^+}.

Proof. This follows from Corollary 5 with rho=|xi|, tau=aleph_xi, and kappa={rm cf}(xi). {sf QED}

Corollary 7. Let xi be an ordinal of uncountable cofinality, and suppose that aleph_alpha^sigma<aleph_{(|xi|^{{rm cf}(xi)})^+} for all cardinals sigma<{rm cf}(xi) and all alpha<xi. Then also aleph_xi^{{rm cf}(xi)}<aleph_{(|xi|^{{rm cf}(xi)})^+}.

Proof. This follows from Corollary 4: If sigma<{rm cf}(xi), then aleph_xi^sigma=aleph_xisup_{alpha<xi}aleph_alpha^sigma, by Theorem II.1.10 from lecture II.2. But xi<(|xi|^{{rm cf}(xi)})^+, so both aleph_xi and sup_{alpha<xi}aleph_alpha^sigma are strictly smaller than aleph_{(|xi|^{{rm cf}(xi)})^+}. {sf QED} 

Corollary 8. If 2^{aleph_alpha}<aleph_{(2^{aleph_1})^+} for all alpha<omega_1, then also  2^{aleph_{omega_1}}<aleph_{(2^{aleph_1})^+}.

Proof. By Corollary 5. {sf QED}

Corollary 9.  If aleph_alpha^{aleph_0}<aleph_{(2^{aleph_1})^+} for all alpha<omega_1, then also  aleph_{omega_1}^{aleph_1}<aleph_{(2^{aleph_1})^+}.

Proof. By Corollary 7. {sf QED}

Notice that, as general as these results are, they do not provide us with a bound for the size of 2^tau for tau the first cardinal of uncountable cofinality that is a fixed point of the aleph sequence, tau=aleph_tau, not even under the assumption that tau is a strong limit cardinal. 

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580 -Cardinal arithmetic (5)

February 13, 2009

At the end of last lecture we defined club sets and showed that the diagonal intersection of club subsets of a regular cardinal is club.

Definition 10. Let \alpha be a limit ordinal of uncountable cofinality. The set S\subseteq\alpha is stationary in \alpha iff S\cap C\ne\emptyset for all club sets C\subseteq\alpha. 

For example, let \lambda be a regular cardinal strictly smaller than {\rm cf}(\alpha). Then S^\alpha_\lambda:=\{\beta<\alpha : {\rm cf}(\beta)=\lambda\} is a stationary set, since it contains the \lambda-th member of the increasing enumeration of any club in \alpha. This shows that whenever {\rm cf}(\alpha)>\omega_1, there are disjoint stationary subsets of \alpha. Below, we show a stronger result. The notion of stationarity is central to most of set theoretic combinatorics. 

Fact 11. Let S be stationary in \alpha.

  1. S is unbounded in \alpha.
  2. Let C be club in \alpha. Then S\cap C is stationary. 

Proof. 1. S must meet \kappa\setminus\alpha for all \alpha and is therefore unbounded.

2. Given any club sets C and D, (S\cap C)\cap D=S\cap(C\cap D)\ne\emptyset, and it follows that S\cap C is stationary. {\sf QED}

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