Math 361, Spring 2015, Assignment 9

Carefully define the following terms, then give one example and one non-example of each:

1. Algebraically closed field.
2. Algebraic closure.
3. Compass-and-straightedge construction.
4. Squaring the circle.
5. Duplicating the cube.
6. Trisection of angles.
7. Constructible number.

Carefully state the following theorems (you do not need to prove them):

1. Theorem giving various conditions characterizing algebraically closed fields.
2. Theorem concerning existence and uniqueness of algebraic closures.
3. Theorem relating constructible numbers to fields.
4. Theorem concerning square roots of constructible numbers.
5. Theorem characterizing constructible numbers in terms of towers of field extensions.
6. Theorem concerning the degree of a constructible number over $\mathbb{Q}$.

Solve the following problems:

1. Section 32, problems 1 and 10.
2. (Objects algebraic over $\mathbb{Z}_p$) Suppose $p$ is prime, that $\mathbb{Z}_p\rightarrow E$ is a field extension, and that $\alpha\in E$ is algebraic over $\mathbb{Z}_p$. Show that the subextension $\mathbb{Z}_p(\alpha)$ has only finitely many elements. (One might say that anything algebraic over $\mathbb{Z}_p$ can be modelled inside a finite field; thus, when we have studied finite fields we will have gone a long way towards understanding the algebraic closure of $\mathbb{Z}_p$.)
3. (Direct limit of a chain of fields) Suppose we have a tower of field extensions $$F_1\rightarrow F_2\rightarrow F_3\rightarrow\dots.$$ We will construct a new field which is in some sense the "union" of all the $F_i$, and which is called the direct limit of the $F_i$. Let $\Omega$ denote the subset of $\mathbb{Z}_{\geq0}\times\bigcup_i F_i$ consisting of all pairs of the form $(i,\alpha)$ with $\alpha\in F_i$. Define a relation $\simeq$ on $\Omega$ by declaring that $(i,\alpha) \simeq (j,\beta)$ if either $i\leq j$ and the image of $\alpha$ in $F_j$ equals $\beta$, or $j\leq i$ and the image of $\beta$ in $F_i$ equals $\alpha$. Show that $\simeq$ is an equivalence relation on $\Omega$, then define operations on $\Omega/\simeq$ that turn it into a field. (This construction can be generalized to non-linear "diagrams" of field extensions. The previous exercise then suggests rather strongly that the algebraic closure of $\mathbb{Z}_p$ can be constructed as the direct limit of all finite fields of characteristic $p$.)

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