Math 361, Spring 2014, Assignment 6

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

1. Constructible number.

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

1. Theorem characterizing the field of constructible numbers.

Solve the following problems:

1. Let $F\rightarrow E$ be any field extension, and suppose $\alpha\in E$. Prove that $\alpha$ is algebraic over $F$ if and only if $\alpha$ lies in some finite-dimensional subextension $K\subseteq E$.
2. Let $\overline{\mathbb{Q}}$ denote the set of all complex numbers that are algebraic over $\mathbb{Q}$. Prove that $\overline{\mathbb{Q}}$ is a subfield of $\mathbb{C}$. (Hint: to show that $\overline{\mathbb{Q}}$ is closed under addition, use the Dimension Formula together with the result of the previous problem. Use similar arguments to demonstrate closure under multiplication and inversion.)
3. Prove that $\overline{\mathbb{Q}}$ is an algebraic closure of $\mathbb{Q}$. (Hint: to show that $\overline{\mathbb{Q}}$ is algebraically closed, suppose we have an algebraic extension $\overline{\mathbb{Q}}\rightarrow E$ and choose $\alpha\in E$. Put $f = \mathrm{irr}(\alpha,\mathbb{Q}) = a_0 + a_1x + \dots + a_nx^n$ and let $L = \mathbb{Q}(a_0,a_1,\dots,a_n, \alpha)$. Apply a strategy similar to that of the previous problem to the extension $\mathbb{Q}\rightarrow L$ to show that in fact $\alpha$ is algebraic over $\mathbb{Q}$. Now consider the factorization of $f$ over $\mathbb{C}$. Argue that this is in fact a factorization of $f$ over $\overline{\mathbb{Q}}$. Finally, conclude that $\alpha$ must lie in (the image of) $\overline{\mathbb{Q}}$.)
4. Describe a compass and straightedge construction that produces an equilateral triangle of side length $1/2$. Describe the construction informally as a sequence of operations with compass and straightedge, as well as formally as a sequence of points having the properties that we described in class.

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Solutions:

Definitions:

1. Constructible Number.

   A real number $\alpha$ is said to be constructible if $|\alpha |$ is the distance between two points occuring in some geometric construction.

   Example:

   $\sqrt{2}$ is constructible.

   Non-example

   $\sqrt[3]{2}$ is not constructible.

Theorems:

1. Theorem characterizing the field of constructible numbers.

   The set of constructible numbers is a subfield of $\mathbb{R}$. Furthermore, a real number $\alpha$ is constructible if and only if there exists a tower of field extensions $$\mathbb{Q}=E_0\subset E_1\subset\dots\subset E_n$$ with $\alpha\in E_n$ and, for each $i$ between $1$ and $n$, $[E_i:E_{i-1}] = 2$.