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{\large \textbf{ Math 31: Topics in Algebra \\ 
                   Summer 2019 - Problem Set 8 }}
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{\hfill Due:   Wednesday, August 21}}
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%\textbf{keywords:} \textit{homomorphism theorem, graph automorphisms} \\
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\textit{Instructions:} Write or type your answers neatly, and staple all pages before passing in your assignment.  Use complete sentences where appropriate.  Show your work; little credit will be given for solutions without work or justification. 

\vspace{1 cm}

\noindent \textbf{\#1} Use Eisenstein's criterion with the appropriate change of variable to show that each of the following polynomials is irreducible over $\mathbb{Q}$:

\begin{enumerate}[a)] 

\item $X^4 + 2X^2 - 1$

\item $X^3 - 3X + 1$

\end{enumerate}

\vspace{0.75cm}

\noindent \textbf{\#2} Use the natural homomorphism $\overline{h}_n: \mathbb{Z}[X] \to \mathbb{Z}_n[X]$ to show that the polynomial $X^4 + 7X^3 + 14X^2 + 3$ has no roots in $\mathbb{Q}$ (find the correct $n$!).  How else might you show there are no rational roots of this polynomial?

\vspace{0.75cm}

\noindent \textbf{\#3}  

\begin{enumerate}[a)]

\item Show that $2 + 3i$ is algebraic over $\mathbb{Q}$ by finding its minimal polynomial $p(X)$.  How do you know $p(X)$ is irreducible?

\item Find a monic irreducible polynomial $p(X)$ over $\mathbb{Q}$ such that $\mathbb{Q}[X]/\langle p(X) \rangle \cong \mathbb{Q}(1 + \sqrt{2})$.

\item Find a complex number $a$ such that $p(X) = X^4 + 2X^2 - 1$ is the minimal polynomial of $a$ over $\mathbb{Q}$.

\end{enumerate}

\vspace{0.75cm}

\noindent \textbf{\#4} Write down the 8 elements of $\mathbb{Z}_2[X]/ \langle X^3 + X + 1 \rangle$.  Write down its multiplication table and confirm every element (except 0) has an inverse.

\vspace{0.75 cm}

\noindent \textbf{\#5} 
\begin{enumerate}[a)]

\item Prove that $\{[a, b, c]^t \, : \, 2a - 3b + c = 0 \}$ is a subspace of $\mathbb{R}^3$.  Find its dimension and a basis for it as a subspace.

\item Prove that if $V$ is a vector space over $F$ with basis $\{\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n}\}$, then for any nonzero scalars $k_1, k_2, \ldots, k_n \in F$, $\{k_1 \overrightarrow{a_1}, k_2 \overrightarrow{a_2}, \ldots, k_n\overrightarrow{a_n}\}$ is also a basis for $V$.

\item Let $U$ and $V$ be vector spaces over $F$ and $h: U \to V$ a linear transformation.  Prove that $\Ker(h)$ is a subspace of $U$ (the \emph{null space}) and $h(U)$ is a subspace of $V$ (the \emph{range space}).

\end{enumerate}

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