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\begin{document}

\begin{center}
{\bf Math 514a: Algebraic Number Theory / 
Fall 2025\\
Homework assignment \#1\\
Suggested due date: September 16, 2025
}
\end{center}

\vspace{5pt}
\color{red} Edit 9/16/25: Added hint for \eqref{unit}. \color{black}


This homework set is a bit of a choose-your-own-adventure. There are three types of problems: A, B, and C. 
\begin{itemize}[itemsep = 3pt]
\item \col{\bf Problems of type A are in blue.} These should be completely straightforward and/or review. Definitely do them if this material is new to you! On the other hand, if one of these is completely obvious to you, it's ok to skip it. 
\item {\bf Problems of type  B are in black.} These are a little more meaty. These are the main problems for the course, and everyone should work on them. 
\item \colred{\bf Problems of type C are in purple.} These are either more challenging or a little outside the main scope of the course. Do these last, if you're all set with A and have already done B. 
\end{itemize}


(Open to suggestions of other forms of typesetting for differentiation; certainly do let me know if the colors are difficult to see.)

\begin{enumerate}[itemsep = 20pt, parsep = 5pt, topsep = 15pt, listparindent = 0pt]

\begin{colortext}

\item \label{legendre} {\bf Legendre symbol review:} Fix an odd prime $p$. For an integer $a$ recall that we set $$\leg{a}{p} := \begin{cases} 0 & \mbox{if $p \mid a$,}\\
1 & \mbox{if $p$ is a nonzero square modulo $a$,}\\
-1 & \mbox{if $p$ is a nonsquare modulo $a$.}
\end{cases}$$
\begin{enumerate}[itemsep = 10pt, topsep = 5pt]
\item In class we showed that $\sleg{a}{p} \equiv a^{\frac{p-1}{2}}$ modulo $p$. Recall this argument. In particular, this implies that $\sleg{-1}{p} \equiv 1$ if and only if $p \equiv 1 \mod{4}$.

More generally, for $d \mid (p-1)$ show that $a^{\frac{p-1}{d}} \equiv 1$ if and only if $a$ is a $d^{\rm th}$ power modulo $p$.  How many distinct $d^{\rm th}$ powers are there modulo $p$?\\ Same question if $\gcd(d, p-1) = 1$.

\item Show that $\sleg{\cdot}{p}$ is a totally multiplicative function $\ZZ \to \{0, 1, -1\}$. That is, $\sleg{a}{p}\sleg{b}{p} = \sleg{ab}{p}$ for all $a, b \in \ZZ$.

\vspace{7pt}

 Show that $\sleg{\cdot}{p}$ is a quadratic Dirichlet character modulo $p$ (look up definitions if necessary). 
\end{enumerate}

\end{colortext}

\noindent To compute $\sleg{a}{p}$ quickly, by multiplicativity it suffices to know $\sleg{2}{p}$ and \emph{quadratic reciprocity:} for odd primes~$p,~q$, we have $\sleg{p}{q} \sleg{q}{p} = (-1)^\frac{p-1}{2}(-1)^\frac{q-1}{2}$.  Both of these were studied by Gauss and have elementary proofs. Proceed as follows. 
\begin{redtext}
\begin{enumerate}[itemsep = 5pt, parsep = 3pt, resume]
\item\label{gausslemma} {\bf Gauss's Lemma:} Given $a$ relatively prime to $p$ and each \mbox{$k = 1, 2, \ldots, \frac{p-1}{2}$} define $r_k \in \{1, \ldots, \frac{p-1}{2}\}$ and $\eps_k = \pm 1$ by $a k  \equiv \eps_k r_k$ modulo $p$. Show that $\sleg{a}{p} = (-1)^n$, where $n$ is the number of $\eps_1, \ldots, \eps_\frac{p-1}{2}$ that are negative. 

 (\emph{Hint}: 
Show that the $r_k$ are distinct and 
%that $r_1, \ldots, r_\frac{p-1}{2}$ are all distinct and c
compare $\prod_k a k$ to $\prod_k \eps_k r_k$.) 


\item \label{gausslemma2} Show that $\eps_k = (-1)^{\left\lfloor\frac{2 a k}{p}\right\rfloor}$ in \eqref{gausslemma} above\footnote{More generally, given positive $a, b$, with $b$ odd, write $a = bq + \eps r$ with $0 \leq r < \frac{b}2$ and $\eps = \pm 1$.\\ Then $\eps = (-1)^{\left\lfloor \frac{2a}{b} \right\rfloor}.$}, so that for $a$ prime to $p$ we have $\sleg{a}{p} = (-1)^m$, where $m = {\sum_{k=1}^{\frac{p-1}{2}} \lfloor\frac{2 a k }{p}\rfloor}.$


\item \label{2onp} 

Show that $\sleg{2}{p} = 1$ iff $p \equiv \pm 1$ modulo $8$.  In other words, $\sleg{2}{p}   = (-1)^\frac{p^2 - 1}{8}.$

\item Show that for odd, positive, relatively prime $a, b$, we have $$\sum_{k=1}^{\frac{b-1}{2}} \left\lfloor\frac{a k }{b}\right\rfloor + \sum_{k=1}^{\frac{a-1}{2}} \left\lfloor\frac{b k }{a}\right\rfloor = \frac{a-1}{2} \cdot \frac{b-1}{2}.$$
(\emph{Hint}: Consider the rectangle in the plane with sides parallel to the axes and one diagonal going from the origin to $(\frac{a}{2}, \frac{b}{2})$. Count the interior lattice points.)

\item Show that for odd, positive, relatively prime $a, b$, the parity of  $\sum_{k=1}^{\frac{b-1}{2}} \left\lfloor\frac{a k }{b}\right\rfloor$ is the same as the parity of  $\sum_{k=1}^{\frac{b-1}{2}} \left\lfloor\frac{2a k }{b}\right\rfloor$. 

Alternatively, for $a$ odd and prime to $p$, use the fact that \mbox{$\textstyle \sleg{2}{p} \sleg{a}{p} = \sleg{2a + 2p}{p} =  \leg{(a+p)/2}{p}$} along with \eqref{gausslemma2} and \eqref{2onp} 
to get rid of the $2$ in \eqref{gausslemma2} and conclude that \mbox{$\sleg{a}{p} = (-1)^M$}, where $M=\sum_{k=1}^{\frac{p-1}{2}} \left\lfloor\frac{a k }{p}\right\rfloor$.

\item {\bf Quadratic reciprocity:} for distinct odd primes~$p,~q$, we have $$\leg{p}{q} \leg{q}{p} = (-1)^\frac{p-1}{2}(-1)^\frac{q-1}{2}.$$ In other words, $\sleg{p}{q} \neq \sleg{q}{p}$ iff $p \equiv q \equiv 3 \mod{4}$. 


\end{enumerate}
\end{redtext}

We will give alternate proofs of both the $\sleg{2}{p}$ statement and quadratic reciprocity this semester by considering prime splitting in cyclotomic fields. 


\begin{colortext}
\item {\bf A bit of algebra review:} Let $A$ be a domain. 

Recall that a nonzero, nonunit element $a$ of $A$ is an \emph{irreducible} element if any factorization $a = bc$ in $A$ if trivial: that is, one of $b$, $c$ is a unit. Under the same assumptions~$a$ is a \emph{prime} element if $a \mid bc$ for $b,c \in A$ implies that $a \mid b$ or $a \mid c$.

\begin{enumerate}[itemsep = 4pt]
\item Show that $a \in A$ is prime if and only if $(a)$ is a nonzero prime ideal of $A$. 
\item Show that $a \in A$ is  irreducible if and only if $(a)$ is maximal among nonzero proper principal ideals of $A$.

\item Show that prime elements are always irreducible. 

\item Give an example of a domain $A$ and an irreducible of $A$ that is not prime.
\end{enumerate}

Recall that  $A$ is \emph{noetherian} if every ideal is finitely generated (equivalently, if every nondecreasing chain of ideals in $A$ eventually stabilizes), and $A$ is a \emph{UFD (unique factorization domain)} if every nonzero nonunit element of $A$ factors into a product of irreducible elements, uniquely up to units and reordering.   

\begin{enumerate}[resume, itemsep = 4pt]

\item Show that in a UFD, irreducible elements are prime. 
\end{enumerate}
\end{colortext}


In fact, irreducible $\iff$ prime close to characterizes a UFD.


\begin{enumerate}[resume, itemsep = 4pt]
\addtocounter{enumii}{5}
\item Show that if $A$ is noetherian, then every nonzero nonunit factors as a product of irreducibles. 

\item Show that a noetherian domain where every irreducible is prime is a UFD.  
\end{enumerate}

\begin{colortext}
More definitions around the chain of reasoning: $A$ is a \emph{PID (principal ideal domain)} if every ideal of $A$ is principal, and $A$ is a \emph{Euclidean domain} if there exists a function $d: A\backslash\{0\} \longrightarrow  \ZZ_{\geq 0}$ so that for every $a, b \in A$ with $b \neq 0$ there is $q, r \in A$ with $a = b q + r$ and $r = 0$ or $d(r) < d(b).$\footnote{To specify the function $d$, we might also say that $A$ is \emph{$d$-Euclidean}.} 

\begin{enumerate}[resume, itemsep = 4pt, parsep = 3pt]
\item \label{euclidPID} Show that if $A$ is Euclidean then it is a PID.

You may assume that $d$ additionally satisfies $d(a) \leq d(ab)$ for all nonzero $b \in A$. Given an ideal $\aa$ of $A$ take an element~$g$ of $J$ with $d(g)$ minimal, and show that~$g$ generates $\aa$.

\colred{More generally, show that you can replace the Euclidean function $d$ by $\tilde d$ which satisfies the additional condition above. For example, set \mbox{$\tilde d(a) := \min_{b \neq 0} d(ab)$} and prove that $A$ is still $\tilde d$-Euclidean.} 
 
\end{enumerate}

The converse to the statement in \eqref{euclidPID} is false: see Theorem 22 of \href{https://kconrad.math.uconn.edu/blurbs/ringtheory/euclideanrk.pdf}{Keith Conrad's blurb on Euclidean domains} for an example of a PID that is not a Euclidean domain.  
\begin{enumerate}[resume, itemsep = 5pt]


\item \label{PIDUFD} Show that if $A$ is a PID then it is a UFD. 

\item The converse to \eqref{PIDUFD} is also false. Give an example of a UFD that is not a PID. 
\end{enumerate}

\end{colortext}

\item {\bf Arithmetic of Gaussian integers:} Recall that $\ZZ[i] = \{a + bi \in \CC: a, b \in \ZZ\}$ with norm function $N: \ZZ[i] \to \ZZ_{\geq 0}$ defined by $N(a + bi) = a^2 + b^2 = |a + bi|_{\CC}^2$. 
\begin{enumerate}[parsep = 5pt, itemsep = 3pt]
\begin{colortext}

\item \label{HG} Show that $N$ is totally multiplicative: $N(\alpha \beta) = N(\alpha) N(\beta)$ for all $\alpha, \beta \in \ZZ[i]$.
\item Show that $\alpha \in \ZZ[i]$ is a unit if and only if $N(\alpha) = 1$.\\ Conclude that the only units in $\ZZ[i]$ are $\pm 1, \pm i$.
\item Show that if $N(\pi)$ is prime in $\ZZ$, then $\pi$ is prime in $\ZZ[i]$.  
\item Show that if $p$ a prime of $\ZZ$ factors in $\ZZ[i]$, then $p = N(\pi)$ for some $\pi \in \ZZ[i]$. 

\item For $\alpha = a + bi$, let $\bar \alpha = a - bi$ be the \emph{conjugate} of $\alpha$. Show that $\alpha$ and $\bar \alpha$ are not associates unless $\alpha$ is in $\ZZ$ or $\alpha$ is of the form $\pm a \pm ai$ for some $a \in \ZZ$ (that is, $\alpha$ is an associate of a rational integer multiple of $1 + i$). 
\end{colortext}

\item In class we gave a geometric argument that $\ZZ[i]$ is norm-Euclidean. In fact, we did a little bit more: we showed that, given $\alpha, \beta \in \ZZ[i]$ with $\beta \neq 0$, we can find $q, r \in \ZZ[i]$ with $\alpha = \beta q + r$ and $N(r) \leq \frac{N(\beta)}{2}$. Give an \emph{algebraic} proof of this  statement. 

(One way to start is to divide $\alpha \bar \beta$ by $N\beta = \beta \bar \beta$, componentwise in $\ZZ$.)

\item In class we showed that a prime $p$ is expressible as a sum of two squares if and only if $p \equiv 1$ modulo $4$. Use factorization in $\ZZ[i]$ to determine which natural numbers are expressible as sums of two squares. 
\end{enumerate}




\item {\bf Generalizing the Gaussian integer story:} First, consider $\ZZ[\sqrt{-2}]$.
\begin{enumerate}[itemsep = 5pt]
\item Define a totally multiplicative norm on $\ZZ[\sqrt{-2}]$. Identify the units of $\ZZ[\sqrt{-2}]$.  
\item Show that $\ZZ[\sqrt{-2}]$ is norm-Euclidean (you can do this either algebraically or geometrically), and hence a PID.
\item Show that $\sleg{-2}{p} = 1$ iff $p \equiv 1, 3$ modulo $8$. (Use \eqref{2onp}.)

\item Give a full description of how primes of $\ZZ$ factor in $\ZZ[\sqrt{-2}]$. Explain everything.

\item Give a full characterization of which positive primes of $\ZZ$ are represented by the quadratic form $x^2 + 2y^2$. 
\end{enumerate}


Now, consider $\ZZ[\frac{1 + \sqrt{-3}}{2}]$. 
\begin{enumerate}[resume]
\item Define a norm, identify the units, and show that $\ZZ[\frac{1 + \sqrt{-3}}{2}]$ is norm-Euclidean.
\item Describe fully how primes of $\ZZ$ factor in $\ZZ[\frac{1 + \sqrt{-3}}{2}]$. Connect this to representations of primes by some quadratic form. 
\end{enumerate}





\item {\bf A real quadratic field:} Now consider $\ZZ[\sqrt{2}] = \{a + b \sqrt{2} \in \RR: a, b \in \ZZ\}$. 
\begin{enumerate}[itemsep = 5pt]
\item Define a totally multiplicative norm $N$ on $\ZZ[\sqrt{2}]$. What can you say about the norm of a unit of $\ZZ[\sqrt{2}]$? 
\item \label{unit} Show that every unit in $\ZZ[\sqrt{2}]$ is of the form $\pm (1 + \sqrt{2})^n$ for some $n \in \ZZ$. In particular, the group of units is a $\ZZ$-module of rank $1$.

\color{red} Edit 9/16/25: \emph{Hint:} If there's a unit in $\ZZ[\sqrt{2}]^\times$ that's not in $U = \{\pm (1 + \sqrt{2})^n\}$, scale it by elements of $U$ to find a unit of the form $a + b \sqrt{2}$ with 

\vspace{-12pt}

 $$1 < a + b \sqrt{2} < 1 + \sqrt{2}.$$ 
 
 \vspace{-3pt} 
 
Invert to get a second inequality; add or subtract to bound $a$ or $b$.

 It may be useful to take cases depending on the sign of the norm.
\color{black}

\item Show that $\ZZ[\sqrt{2}]$ is norm-Euclidean and hence a PID.  

\item How do primes of $\ZZ$ factor in $\ZZ[\sqrt{2}]$? Which primes of $\ZZ$ are represented by the quadratic form $x^2 - 2y^2$? 
\end{enumerate}


\item {\bf Rings of integers in quadratic fields:} Let $K$ be a quadratic extension of $\QQ$. Compute its ring of integers $\OO_K$. (First show that $K = \QQ(\sqrt{d})$ for some squarefree integer $d$; your answer will depend on $d$ modulo $4$, and you should find three cases.)

\vfill

\item {\bf Minkowski's theorem:} A region $D$ of $\RR^n$ is said to be \emph{convex} if whenever two points $P, Q$ are in $D$, the entire line segment $\{t P + (1-t)Q: t \in [0, 1]\}$ connecting $P$ and $Q$ is contained in $D$ as well.\footnote{I believe I said in class that is suffices to check midpoints, but that's false without additional assumptions: indeed, $\QQ \subset \RR$ is ``midpoint-closed" but is not a convex set!} We will call a region $D$ of $\RR^n$ (\emph{centrally}) \emph{symmetric} if $-P$ is in $D$ whenever $P$ is in $D$. 

A \emph{lattice} in $\RR^n$ here will be a free $\ZZ$-module of rank $n$ whose basis is an $\RR$-basis for~$\RR^n$. The covolume $\Covol L$ of a lattice $L$ is the volume of a \emph{fundamental domain} of the lattice (the $n$-parallelepiped determined by a basis of the lattice). 
\begin{enumerate}[itemsep = 5pt, parsep= 3pt]
\item Prove {\bf Minkowski's theorem}: Let $L$ be a lattice in $\RR^n$. If $D \subseteq \RR^n$ is a convex, symmetric, measurable region with $\Vol(D) > 2^n \Covol(L)$, then $D$ contains a nonzero point of $L$. 

(First show that it suffices to assume that $L$ is the standard lattice $\ZZ^n$ of $\RR^n$. Translating $D$ into a fundamental domain of the doubled lattice $2 \ZZ^n$, show that volume considerations tell us that $D$ contains two distinct points $P$, $Q$ all whose coordinates differ by even integers.)
\item {\bf Four-square theorem for primes:} Use Minkowski's theorem to prove Lagrange's result that any prime is expressible as a sum of \emph{four} squares. 

(If~$p$ is an odd prime, show that one can find two integers $a, b$ so that $a^2 + b^2 + 1 \equiv 0$ modulo $p$ (pigeonhole principle). Consider the region $$D = \{(x, y, z, w): x^2 + y^2 + z^2 + w^2 < 2p\} \subset \RR^4$$ and the lattice $L \subset \RR^n$ spanned by $(p, 0, 0, 0)$, $(0, p, 0, 0)$, $(a, b, 1, 0)$, and $(-b, a, 0, 1)$. You'll have to derive or look up a formula for the volume of a $4$-sphere.) 

\begin{redtext}
\item Prove Lagrange's full result: every nonnegative integer is expressible as a sum of four squares. 

(One convenient way to proceed is to put a multiplicative norm on the Hamiltonian quaternions $\HH$. If $\alpha = a + bi + cj + dk \in \HH$, let $\bar \alpha := a - bi - cj - dk$ and set $N\alpha := \alpha \bar \alpha = a^2 + b^2 + c^2 + d^2$. Check that $\overline{\alpha \beta} = \bar \beta \bar \alpha$ to establish multiplicativity. Restrict to $a, b, c, d \in \ZZ$.)
\end{redtext}
\end{enumerate}
\begin{colortext}
\item {\bf Transitivity of norm and trace:} Let $M/L/K$ be a finite tower of fields. 
\begin{enumerate}
\item Show that $\N_{M/K} = \N_{L/K} \circ \N_{M/L}$.  
\item Show that $\Tr_{M/K} = \Tr_{L/K} \circ \Tr_{M/L}$.  
\end{enumerate}
\end{colortext}


\end{enumerate} 
\end{document}