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\title{Homework 6: Due Friday March 2nd 2018\\
MTH 310 001 W18}
\author{YOUR NAME HERE}
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Solutions must be typed using LaTeX.
Remember to provide justification and write in complete sentences for all responses unless otherwise noted. \\
\begin{enumerate}
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% Question 1
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\item Sec. 4.1. Problem 4. In each part, give an example polynomials $f(x),g(x)\in\bQ[x]$ that satisfy the condition:
\begin{enumerate}
\item The degree of the sum of $f(x)$ and $g(x)$ is less than the maximum of $\text{deg}(f(x))$ and $\text{deg}(g(x))$; i.e. \[\text{deg}(f(x)+g(x))<\text{max}\{ \text{deg}(f(x)),\text{deg}(g(x))\}.\]
\item The degree of the sum of $f(x)$ and $g(x)$ is equal to the maximum of $\text{deg}(f(x))$ and $\text{deg}(g(x))$; i.e. \[\text{deg}(f(x)+g(x))=\text{max}\{ \text{deg}(f(x)),\text{deg}(g(x))\}.\]
\end{enumerate}
\begin{proof}
Type your answer here!
\end{proof}
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% Question 2
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\item Sec. 4.1. Problem 8. If $R$ has a multiplicative identity $1_R$, show that the constant polynomial $1_R$ is the multiplicative identity of $R[x]$.
\begin{proof}
Type your answer here!
\end{proof}
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% Question 3
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\item Let $R$ be an integral domain and let $R^{\times}$ be the units in $R$. Prove that the only units in $R[x]$ are the constant polynomials $b$ where $b$ is an element in $R^{\times}$.
\begin{proof}
Type your answer here!
\end{proof}
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% Question 4
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\item Show that $1+3x$ is a unit in $\mathbb{Z}/9\mathbb{Z}[x]$. State why this doesn't contradict the previous problem.
\begin{proof}
Type your answer here!
\end{proof}
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% Question 5
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\item Sec. 4.1. Problem 18. Let $\psi:R[x]\rightarrow R$ be the function that maps $f(x)=a_0+a_1x+\dots a_nx^n$ to $a_0$. Prove that $\psi$ is a homomorphism. Prove or disprove that it is surjective. Prove or disprove that it is an isomorphism of rings.
\begin{proof}
Type your answer here!
\end{proof}
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% Question 6
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\item Sec. 4.2. Problem 1. Let $F$ be a field. If $f(x)\in F(x)$, show that every nonzero constant polynomial divides $f(x)$.
\begin{proof}
Type your answer here!
\end{proof}
\end{enumerate}
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