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\date{Due: December 8, 2025}

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\noindent \textbf{Homework Assignment 4\hfill Math 369 - Introduction to Abstract Algebra}
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\noindent\textit{``In mathematics you don't understand things. You just get used to them.''}   \hfill --- John von Neumann
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\noindent \textbf{Instructions:} Show all work clearly. Justify your answers with complete explanations. Late assignments will not be accepted without prior arrangement.

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\noindent \textbf{Problem 1.} Let $n \geq 2$ be an integer.

\begin{enumerate}
\item[(a)] Prove that if $\phi: \mathbb{Z} \to \mathbb{Z}_n$ is a ring homomorphism, then $\phi(k) = [k]_n$ for all $k \in \mathbb{Z}$. (That is, the only ring homomorphism from $\mathbb{Z}$ to $\mathbb{Z}_n$ is the natural projection.)
\item[(b)] More generally, let $\phi: \mathbb{Z} \to R$ be a ring homomorphism where $R$ is any commutative ring. Show that $\phi$ is completely determined by $\text{char}(R)$.
\end{enumerate}

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\noindent \textbf{Problem 2.} Let $F_1$ and $F_2$ be fields. The direct sum $F_1 \oplus F_2$ consists of all ordered pairs $(a, b)$ with $a \in F_1$ and $b \in F_2$, under componentwise operations:
$$(a_1, b_1) + (a_2, b_2) = (a_1 + a_2, b_1 + b_2) \quad \text{and} \quad (a_1, b_1) \cdot (a_2, b_2) = (a_1a_2, b_1b_2)$$

\begin{enumerate}
\item[(a)] Show that $F_1 \oplus F_2$ is a commutative ring with identity.
\item[(b)] Prove that $F_1 \oplus F_2$ is not a field.
\end{enumerate}

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\noindent \textbf{Problem 3.} Let $R = \mathbb{Z}[i] = \{a + bi : a, b \in \mathbb{Z}\}$ be the ring of Gaussian integers, and let $I = \langle 5 \rangle$ be the principal ideal generated by $5$.

\begin{enumerate}
\item[(a)] Show that $5 = (1+2i)(1-2i)$ in $\mathbb{Z}[i]$.
\item[(b)] Use part (a) to prove that $I$ is not a prime ideal.
\item[(c)] Determine whether $I$ is a maximal ideal in $\mathbb{Z}[i]$.
\end{enumerate}

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\noindent \textbf{Problem 4.} Let $R$ be an integral domain of characteristic $p > 0$. The \emph{Frobenius homomorphism} is the map $\phi: R \to R$ defined by $\phi(a) = a^p$.

\begin{enumerate}
\item[(a)] Prove that $\phi$ is a ring homomorphism. (Hint: You will need the Binomial Theorem and the fact that $p$ divides $\binom{p}{k}$ for $0 < k < p$.)
\item[(b)] Show that if $R$ is a finite field, then $\phi$ is an isomorphism.
\end{enumerate}

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\noindent \textbf{Problem 5.} Let $R$ be a commutative ring and let $I, J$ be ideals of $R$ with $I + J = R$ (meaning there exist $a \in I$ and $b \in J$ such that $a + b = 1$).

\begin{enumerate}
\item[(a)] Prove that for any $r \in R$, there exist $x \in I$ and $y \in J$ such that $r = x + y$.
\item[(b)] Define $\phi: R \to R/I \times R/J$ by $\phi(r) = (r + I, r + J)$. Prove that $\phi$ is a ring homomorphism with $\ker(\phi) = I \cap J$.
\item[(c)] Use the Fundamental Homomorphism Theorem to conclude that $R/(I \cap J) \cong R/I \times R/J$.
\end{enumerate}

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