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\newcommand{\courseName}{Math 480A2}
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{\large \courseName, Homework \hwNumber}\\
Due \hwDueDate\\
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{\small \textit{Homework is graded out of a total of 10 points. Collaboration is permitted, but you must list all coauthors on a problem's solution at the top of the page, and your writing must be your own.}}
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\textbf{Problem 1.} (3 points) Describe the elements which are units in the ring $R = \ZZ / 12 \ZZ$. List the ideals of $R$ (express each as a set), and explain why each set is an ideal, and why every ideal is accounted for.
\vspace{3mm}
\textbf{Problem 2.} (2 points) Let $R, R'$ be commutative rings, let $\varphi: R \to R'$ be a homomorphism, and let $I'$ be an ideal in $R'$. Prove that the preimage
\[
\varphi^{-1}(I') = \{r \in R : \varphi(r) \in I' \}
\]
is an ideal in $R$.
\vspace{3mm}
\textbf{Problem 3.} (2 points) Let $f(x) = 3x^5 + x^4 + 5x^3 + 4x^2 + 4x + 6$ and let $g(x) = x^3 + x + 1$ be polynomials in $\ZZ[x]$. Use polynomial long division to find quotient and remainder polynomials $q$ and $r$ such that $f(x) = q(x) g(x) + r(x)$, where $r$ has degree less than the degree of $g$.
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\textbf{Problem 4.} (3 points) Let $R = \ZZ / 2 \ZZ$, and let $F = R[x] / (x^2 + x + 1)$, where $x^2 + x + 1$ is interpreted as a polynomial with coefficients in $\ZZ / 2 \ZZ$. How many elements does $F$ have? (Hint: use polynomial long division to express each element of $F$ in a simple ``canonical'' form.) Write out the addition and multiplication table of $F$, and explain why $F$ is a field.
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