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\begin{document}
\assignment{1}
% Problem 1
\problem Show that there exists no nontrivial unramified extensions of $\Q$. \pspace
\solution If $K/\Q$ is a nontrivial number field, then $|\disc K|>1$. But then $\disc K$ has a prime factor so that some prime ramifies in $K$. \qed \pspace
% Problem 2
\problem Complete the following:
\begin{enumerate}[(a)]
\item How does one prove a cotheorem?
\item Compute $\ds \int \cos x \;dx$.
\item How does one square $\twomatrix{a}{b}{c}{d}$?
\end{enumerate}
\solution
\begin{enumerate}[(a)]
\item Use rollaries.
\item We have
\begin{equation} \label{eq:integral}
\int \cos x \;dx= \sin x + C
\end{equation}
We can check \eqref{eq:integral}:
\[
\dfrac{d}{dx} \left( \sin x + C \right)= \cos x
\]
\item This is routine.
\end{enumerate} \qed \pspace
% Problem 3
\problem Prove that $\sqrt{2}$ is irrational. \pspace
\pf Assume that $\sqrt{2}= \dfrac{a}{b}$, where $a,b \in \Z$. Without loss of generality, we may assume $\gcd(a,b)= 1$. Then we have
\begin{align}
\sqrt{2}&= \dfrac{a}{b} \nonumber \\
\sqrt{2}^2&= \left( \dfrac{a}{b} \right)^2 \label{eq:implication1} \\
2&= \dfrac{a^2}{b^2} \nonumber \\
a^2&= 2b^2 \label{eq:implication2}
\end{align}
But then from \eqref{eq:implication2}, we know that $a^2$ is even so that $a$ is even. But then we must have
\[
2a^2= b^2
\]
so that $b^2$ is even, implying $b$ is even. But then $\gcd(a,b) \geq 2$, a contradiction. \qed \\
\end{document}
