% HMC Math dept HW template example
% v0.04 by Eric J. Malm, 10 Mar 2005
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\name{}
\class{Math 315}
\assignment{Homework \#3}
\duedate{4/14/2017}

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\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\II}{\mathbb{I}}

\begin{document}

\problemlist{Abbott: 1.4.5, 1.5.9}

\begin{problem}[1.4.5]
Using Exercise 1.4.1, supplly a proof for Corollary 1.4.4 by considering the real numbers $a - \sqrt{2}$ and $b - \sqrt{2}$.
\end{problem}

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\begin{problem}[1.5.9]
A real number $x \in \RR$ is called \emph{algebraic} if there exist integers $a_0, a_1, a_2, \ldots, a_n \in \ZZ$, not all zero, such that
\[ a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0. \]
Said another way, a real number is algebraic if it is the root of a polynomial with integer coefficients. Real numbers that are not algebraic are called \emph{transcendental} numbers. Reread the last paragraph of Section 1.1. The final question posed here is closely related to the question of whether or not transcendental numbers exist.
\begin{enumerate}
\item Show that $\sqrt{2}$, $\sqrt[3]{2}$, and $\sqrt{3} + \sqrt{2}$ are algebraic.
\item Fix $n \in \NN$, and let $A_n$ be the algebraic numbers obtained as roots of polynomials with integer coefficients that have degree $n$. Using the fact that every polynomial has a finite number of roots, show that $A_n$ is countable.
\item Now, argue that the set of all algebraic numbers is countable. What may we conclude about the set of transcendental numbers?
\end{enumerate}
\end{problem}

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