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\newcommand{\courseName}{Math 480A2}
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\newcommand{\hwDueDate}{September 15, 2022}


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\begin{document}

\begin{center}
	{\large \courseName, Homework \hwNumber}\\
	Due \hwDueDate\\
\end{center}

{\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.}}

\vspace{5mm}

\textbf{Problem 1.} (3 points) Compute the monic irreducible polynomials of degree 4 over the binary field $F_2 = \ZZ / 2 \ZZ$.  (\textit{Hint:} if a polynomial of degree 4 is reducible, it either has a linear factor and thus a root in $F_2$, or it can be written as a product of monic irreducible polynomials of degree 2.)

\vspace{3mm}

\textbf{Problem 2.} (2 points) Let $F_3 = \ZZ / 3 \ZZ$, and let $f(x) = x^4 + x - 1 \in F_3[x]$.  Then $f$ is irreducible over $F_3$, so the quotient $K = F_3[x] / (x^4 + x - 1)$ is a field.  Compute the inverse of $x^2 + 1$ in $K$.

\textbf{Challenge.} (1 bonus point) Prove that $x^4 + x - 1$ is irreducible over $F_3$.

\vspace{3mm}

\textbf{Problem 3.} (3 points) Prove that the following numbers are algebraic over $\QQ$ by finding a nonzero polynomial with rational coefficients for which they are a root: $1 + \sqrt{2}$, $\sqrt{1 + \sqrt{2}}$, $\sqrt{2} + \sqrt{3}$.

\vspace{3mm}

\textbf{Problem 4.} (2 points) Compute (with proof) the irreducible polynomial of the golden ratio $\phi = (1 + \sqrt{5}) / 2$ over $\QQ$.

\end{document}