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 where is a polynomial with complex coefficients, has at least one complex root This means that is a complex number and Show that has at most roots, where is its degree, and that if we count roots up to multiplicity, then it has exactly roots. Since the multiplicity of a root is by definition the largest such that is a factor of you may want to verify that iff is a factor of

2. Let be a polynomial with real coefficients, and let be a complex root of Show that as well. Conclude that if the degree of is odd and the coefficients of are real, then has at least one real root. (You may use the fundamental theorem of algebra, if needed.) Conclude also that if is of degree four and has real coefficients, then 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 are real numbers, then at least one of the solutions of 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).

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Stefan, "low" cardinalities do not change by passing from $L({\mathbb R})$ to $L({\mathbb R})[{\mathcal U}]$, so the answer to the second question is that the existence of a nonprincipal ultrafilter does not imply the existence of a Vitali set. More precisely: Assume determinacy in $L({\mathbb R})$. Then $2^\omega/E_0$ is a successor cardinal to ${ […]

Marginalia to a theorem of Silver (see also this link) by Keith I. Devlin and R. B. Jensen, 1975. A humble title and yet, undoubtedly, one of the most important papers of all time in set theory.

Given a positive integer $a$, the Ramsey number $R(a)$ is the least $n$ such that whenever the edges of the complete graph $K_n$ are colored using only two colors, we necessarily have a copy of $K_a$ with all its edges of the same color. For example, $R(3)= 6$, which is usually stated by saying that in a party of 6 people, necessarily there are 3 that know e […]

No, this is not consistent. Todorčević has shown in ZF that, in fact, there is no function $F\!:\mathcal W(S)\to S$ with the property you require. Here, $\mathcal W(S)$ is the collection of subsets of $S$ that are well-orderable. This is corollary 6 in MR0793235 (87d:03126). Todorčević, Stevo. Partition relations for partially ordered sets. Acta Math. 155 (1 […]

As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as their "weak product"). The general notion is a clear generalization, although the original reference only deals with groups. As mentioned by Cameron Zwa […]

Gödel sentences are by construction $\Pi^0_1$ statements, that is, they have the form "for all $n$ ...", where ... is a recursive statement (think "a statement that a computer can decide"). For instance, the typical Gödel sentence for a system $T$ coming from the second incompleteness theorem says that "for all $n$ that code a proof […]

When I first saw the question, I remembered there was a proof on MO using Ramsey theory, but couldn't remember how the argument went, so I came up with the following, that I first posted as a comment: A cute proof using Schur's theorem: Fix $a$ in your semigroup $S$, and color $n$ and $m$ with the same color whenever $a^n=a^m$. By Schur's theo […]

It depends on what you are doing. I assume by lower level you really mean high level, or general, or 2-digit class. In that case, 54 is general topology, 26 is real functions, 03 is mathematical logic and foundations. "Point-set topology" most likely refers to the stuff in 54, or to the theory of Baire functions, as in 26A21, or to descriptive set […]

In the presence of the axiom of foundation, it is true as you indicate that no set belongs to itself, and so the definition of transitive set can be written with $\subset$ (or $\subsetneq$, whichever symbol you prefer). However, one may study also set theories where foundation fails, and then it is natural to define transitive sets in a way that allows self- […]

You do not need much to recover the full ultrapower. In fact, the $\Sigma_1$-weak Skolem hull should suffice, where the latter is defined by using not all Skolem functions but only those for $\Sigma_1$-formulas, and not even that, but only those functions defined as follows: given a $\Sigma_1$ formula $\varphi(t,y_1,\dots,y_n)$, let $f_\varphi:{}^nN\to N$ be […]

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