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\large
\textbf{18.785 Number Theory\hspace{228pt}Fall~2025}\\\vspace{4pt}
\textbf{Problem Set \#3\due{09/26/2025}}\\\vspace{-6pt}
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\noindent
These problems are related to material in Lectures 4--6.
Your solutions should be written up in latex and submitted as a pdf-file \ifOCW\else to \href{https://www.gradescope.com/courses/283349}{Gradescope} \fi by midnight on the date due.
\medskip

\noindent
\textbf{Instructions:} Solve Problem 0, then pick any combination of problems that sum to 100 points. Collaboration is permitted/encouraged, but you must identify your collaborators (including any LLMs you consulted) and any references you consulted outside the course \ifOCW syllabus \else\href{https://math.mit.edu/classes/18.785/2025/syllabus.html}{syllabus}\fi.
Include this information after the \textbf{Collaborators/Sources} prompt at the end of the problem set  (if there are none, you should enter ``none'', do not leave it blank).
Each student is expected to write their own solutions; it is fine to discuss problems with others, but your writing must be your own.

\subsection*{Problem 0. Warmup (0 points)}
These warmup exercises do not need to be written up.
\begin{enumerate}
\setlength\itemsep{0pt}
\item[\textbf{(a)}] Let $K$ be the field $\Fp(x,y)$ and consider the field $L:=K[t]/(t^{p^2}+t^px+y)$.  Show that $L/K$ can be decomposed as a purely inseparable extension of a separable extension, but not as a separable extension of a purely inseparable extension.
\item[\textbf{(b)}] Show that the \'etale algebra $\F_2^3$ is not isomorphic to $\F_2[x]/(f)$ for any $f\in \F_2[x]$.
\item[\textbf{(c)}] Let $K$ and $L$ be two number fields.  Describe the finite \`etale $K$-algebra $L\otimes_\Q K$ when $L\subseteq K$, $K\subseteq L$, $K=L$, $K\cap L=\Q$, and then in general.
\end{enumerate}

\subsection*{Problem 1. Perfect closures (33 points)}
Let $k$ be a field.  A purely inseparable algebraic extension of $k$ that is perfect is called a \textit{perfect closure} of $k$.
\begin{enumerate}
\item[\textbf{(a)}] Prove that every field has a perfect closure.
\item[\textbf{(b)}] Prove that the perfect closure is unique up to a unique isomorphism: if $L_1$ and $L_2$ are two perfect closures of $k$ then there is a unique $k$-algebra isomorphism $L_1\to L_2$.
\end{enumerate}
In view of \textbf{(b)}, we use $k^{\textrm{perf}}$ to denote the (essentially unique) perfect closure of $k$.
\begin{enumerate}
\item[\textbf{(c)}] Let $L$ be a separable extension of $k$ and let $M$ be a purely inseparable extension of $k$.  Show that the $k$-algebra $L\otimes_k M$ is a field (Hint: Reduce to finite extensions and show that the quotient of $L\otimes_kM$ by a maximal ideal has the same degree).
\item[\textbf{(d)}] Let $\kbar$ be an algebraic closure of $k$, let $k^{\rm sep}$ be the separable closure of $k$ in $\kbar$, and let $k^{\rm perf}$ be the perfect closure of $k$ in $\kbar$. Let $L$ be the $k$-algebra $k^{\rm sep} \otimes_k k^{\rm perf}$, which by \textbf{(c)} is a field.  Show that $L$ is an algebraic extension of both $k^{\rm sep}$ and $k^{\rm perf}$ and can thus be embedded in $\kbar$.  Then show that this embedding is an isomorphism.
\end{enumerate}

\subsection*{Problem 2. Counting irreducible polynomials over finite fields (33 points)}
Let $k$ be a finite field.  Let $q=\#k$.
For $d \ge 1$, let $N_d$ be the number of degree $d$ monic irreducible polynomials in $k[x]$.
\begin{enumerate}[\textbf{(a)}]
\item[\textbf{(a)}] Adapt the proof of the Euler product identity
\[
    \sum_{n \ge 1} n^{-s} = \prod_{\textup{prime $p$}} (1+p^{-s}+p^{-2s}+\cdots)
\]
to prove the function field analogue 
\[
    \sum_{\textup{monic $f \in k[x]$}} x^{\deg f} = \prod_{\textup{monic irreducible $g \in k[x]$}} (1 + x^{\deg g} + x^{2 \deg g} + \cdots).
\]
\item[\textbf{(b)}] Prove $(1-qx)^{-1} = \prod_{d \ge 1} (1-x^d)^{-N_d}$ in $\Z[[x]]$.
\item[\textbf{(c)}] Prove $q^n = \sum_{d|n} d N_d$ for each $n \ge 1$.
\item[\textbf{(d)}] Prove $N_n = \frac{1}{n} \sum_{d|n} \mu(n/d) \, q^d$ for each $n \ge 1$.
\item[\textbf{(e)}] Prove that for every $n \ge 1$, there exists a degree~$n$ monic irreducible $f\in k[x]$.
\item[\textbf{(f)}] Prove that a random monic $f\in k[x]$ of degree $n$ is irreducible with probability $\frac{1}{n} + O(n^{-1} q^{-n/2})$.
\item[\textbf{(g)}] Use the theory of finite fields to give an alternative proof of the formula in \textbf{(c)} by grouping elements of $\F_{q^n}$ by minimal polynomial.
\end{enumerate}

\subsection*{Problem 3. Class groups of quadratic rings (33 points)}
Let $A=\Z[\sqrt{-5}]$.
\begin{enumerate}
\item[\textbf{(a)}] Prove that each ideal class in $A$ is represented by a fractional ideal $I$ in which $1$ is an element of smallest complex absolute value.
\item[\textbf{(b)}] Prove that the only such fractional ideals are $(1)$ and $(1,(1+\sqrt{-5})/2)$.
\item[\textbf{(c)}] Compute the ideal class group $\mathcal \cl(A)$.
\end{enumerate}
Now let $A=\Z[\sqrt{-d}]$ for some positive integer $d$.
\begin{enumerate}
\item[\textbf{(d)}] Prove that there is a constant $B$ depending on $d$, such that any set of more than $B$ points in the rectangle $R:= [0,1] + [0,\sqrt{d}]i$ contains two distinct points separated by a distance less than $1$.
\item[\textbf{(e)}] Prove that if $I$ is a fractional ideal in which $1$ is an element of smallest complex absolute value, then $[I: A] \le B$.
\item[\textbf{(f)}] Prove that for each $n \ge 1$, there are only finitely many fractional ideals $I$ with $A \subseteq I \subseteq \Frac A$ and $[I: A] = n$.
\item[\textbf{(g)}] Prove that $\cl(A)$ is finite.
\end{enumerate}

\subsection*{Problem 4. Factoring primes in quadratic fields (33 points)}
Let $p,q\in \Z$ denote (not necessarily distinct) primes.
\begin{enumerate}
\item[\textbf{(a)}] Let $K$ be a quadratic extension of $\Q$ with ring of integers $\O_K$  As we proved in Lecture 5, $\O_K$ is a Dedekind domain (as are all rings of integers).  Let
\[
(q)=\q_1^{e_1}\cdots\q_n^{e_n}
\]
be the unique factorization of the principal ideal $(q)$ in $\O_K$.  Show that
\[
[\O_K:q\O_K]=q^2=\prod_{i=1}^n [\O_K:\q_i]^{e_i},
\]
(where $[B:A]$ denotes the index of $A$ in $B$ as an additive abelian group),
and conclude that there are three possibilities: $(q)$ is prime, $(q)=\q_1\q_2$, or $(q)=\q_1^2$.
\item[\textbf{(b)}] For $K:=\Q(\sqrt{p})$ determine the unique factorization of $(q)$ in $\O_K$ explicitly; that is, determine which of the three possibilities admitted by \textbf{(a)} occurs and when applicable, write $\q_i$ in the form $(q,\alpha_i)$ for some explicitly described $\alpha\in \O_K$.  Be sure to address the cases $q=2$ and $q=p$ which may require special treatment.
\item[\textbf{(c)}] Do the same for $K:=\Q(\sqrt{-p})$.
\item[\textbf{(d)}] For primes $p,q\ne 2$, let $K:= \Q(\sqrt{\pm p})$ and relate the factorization of $(q)$ in $\O_K$ you determined in parts (b) and (c) to the factorization of $x^2\mp p$ in $\Fq[x]$.
\end{enumerate}

\subsection*{Problem 5. Computing norms and traces (33 points)}
Let $L/K$ be a finite extension of fields, let $\Kbar$ be an algebraic closure containing $L$, and let $\Sigma:=\Hom_K(L,\Kbar)$.
Let $\alpha\in L$ have minimal polynomial $f(x)=\sum_{i=0}^d a_ix^i\in K[x]$, and let $f(x)=\prod_{i=1}^d(x-\alpha_i)$ its factorization in $\Kbar[x]$.
Define $n:=[L:K]$ and $e:=[L:K(\alpha)]$.

\begin{enumerate}
\item[\textbf{(a)}]Prove that
\[
\N_{K(\alpha)/K}(\alpha)=\prod_{i=1}^d\alpha_i=(-1)^d a_0\qquad\text{and}\qquad\T_{K(\alpha)/K}(\alpha)=\sum_{i=1}^d \alpha_i=-a_{d-1}.
\]
(Hint: Consider the companion matrix of $f$).
\item[\textbf{(b)}] Show that if $L/K$ is purely insepararable then
\[
N_{L/K}(\alpha)=\alpha^{[L:K]}\qquad\text{and}\qquad T_{L/K}(\alpha)=[L:K]\alpha = \begin{cases}\alpha & \text{if }L=K\\0 & \text{if }L\ne K\end{cases}.
\]
\item[\textbf{(c)}] Prove that
\[
\N_{L/K}(\alpha)=\prod_{i=1}^d\alpha_i^e=(-1)^na_0^e\qquad\text{and}\qquad\T_{L/K}(\alpha)=\sum_{i=1}^de\alpha_i=-ea_{d-1}.
\]
\item[\textbf{(d)}] Prove that for all $\alpha\in L$ we have
\[
N_{L/K}(\alpha)=\left(\prod_{\sigma\in\Sigma}\sigma(\alpha)\right)^{[L:K]_i}\qquad\text{and}\qquad T_{L/K}(\alpha)=[L:K]_i\left(\sum_{\sigma\in\Sigma}\sigma(\alpha)\right).
\]
\item[\textbf{(e)}] Prove that $T_{L/K}=0$ (as a linear map) if and only if $L/K$ is inseparable.\\
(Hint: Use base change to handle the separable case.)
\end{enumerate}
\smallskip

\subsection*{Problem 6. Survey (1 point)}
Complete the following survey by rating each problem you attempted on a scale of 1 to~10 according to how interesting you found it (1 = ``mind-numbing," 10 = ``mind-blowing"), and how difficult you found it (1 = ``trivial," 10 = ``brutal").  Also estimate the amount of time you spent on each problem to the nearest half hour.

\begin{center}
\begin{tabular}{l|r|r|r|}
& Interest & Difficulty & Time Spent\\\hline
Problem 1 & & & \\\hline
Problem 2 & & & \\\hline
Problem 3 & & & \\\hline
Problem 4 & & & \\\hline
Problem 5 & & & \\\hline
\end{tabular}
\end{center}
\noindent
Please feel free to record any additional comments you have on the problem sets and the lectures, and in particular, ways in which they might be improved.
\medskip

\noindent
\textbf{Collaborators/sources}:

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