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\begin{document}
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\large
\textbf{18.785 Number Theory\hspace{228pt}Fall~2025}\\\vspace{4pt}
\textbf{Problem Set \#7\due{11/3/2025}}\\\vspace{-6pt}
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\subsection*{Description}

These problems are related to Lectures 11--13.
Your solutions should be written up in latex and submitted as a pdf-file \ifOCW\else to \href{https://www.gradescope.com/courses/1110057}{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).
Note that each student is expected to write their own solutions; it is fine to discuss problems with others, but your writing must be your own.
The \href{https://www.lmfdb.ofg}{LMFDB} provides a rich source of examples of number fields and $p$-adic fields that you may find useful (even just for building intuition).  You may cite examples in the LMFDB without further proof.

\subsection*{Problem 0.}
These are warm-up questions that do not need to be turned in.
\begin{enumerate}
\setlength\itemsep{0pt}
\item[(a)] Prove that the absolute discriminant of a number field is always a square mod~$4$.
\item[(b)] Compute the different ideals of the quadratic fields $\Q(\sqrt{-2})$ and $\Q(\sqrt{-3})$.
\item[(c)] Determine the primes that ramify in $\Q[x]/(x^3-x-1)$ and $\Q[x]/(x^3+x+1)$ and compute their ramification indices.
\item[(d)] Let $p$ be an odd prime.  Compute the different ideal and absolute discriminant of the cyclotomic extension $\Q(\zeta_p)/\Q$.
\end{enumerate}

\subsection*{Problem 1. The different ideal (66 points)}

Let $A$ be a Dedekind domain with fraction field $K$, let $L/K$ be a finite separable extension, and let $B$ be the integral closure of $A$ in $L$.
Write $L=K(\alpha)$ with $\alpha\in B$ and let $f\in A[x]$ be the minimal polynomial of $\alpha$, with degree $n=[L:K]$.
\begin{itemize}
\item[(\bf a)] By comparing the formal expansion of $1/f(x)$ at infinity with its partial fraction decomposition over the splitting field of $f$ (the Galois closure of $L$), prove that
\[
\T_{L/K}\left(\frac{\alpha^i}{f'(\alpha)}\right)=\begin{cases}
0&\text{if }0\le i\le n-2;\\
1&\text{if }i=n-1;\\
\in A&\text{if }i\ge n.
\end{cases}
\]

\item[{\bf (b)}] Suppose $B=A[\alpha]$.  Prove that $B^*:=\{x\in L:\T_{L/K}(xb)\in A\text{ for all }b\in B\}$ is the principal fractional $B$-ideal $(1/f'(\alpha))$.  Conclude that $\D_{B/A}=(f'(\alpha))$.

\item[{\bf (c)}]Prove that if $g$ is the minimal polynomial of an element $\beta\in B$ for which $L=K(\beta)$ then $\N_{L/K}(g'(\beta))=\pm \disc(g)$.

\item[{\bf (d)}] By Proposition 12.26 we have $\D_{B/A}=(\delta_{B/A}(\beta)\colon \beta\in B)$, where $\delta_{B/A}(\beta)$ is $g'(\beta)$ if the minimal polynomial $g$ of $\beta$ has degree $n$ and zero otherwise.  Prove or disprove:
\[
D_{B/A}\overset{?}=(\N_{L/K}(\delta_{B/A}(\beta)):\beta\in B).
\]

\item[{\bf (e)}] Now consider on order $\O$ with integral closure $B$ (equivalently, an $A$-lattice in $L$ that is a ring) and conductor $\c$ (as in Definition 6.16).  Define the \emph{different} of $\O$ as
\[
\D_{\O/A} := \{x\in L: x\O^*\subseteq B\},
\]
and define the \emph{discriminant} of $\O$ (as an $A$-lattice in $L$, see Definition 12.9) via
\[
D_{\mathcal O/A}:=D(\mathcal O).
\]
Prove that $\D_{\O/A}=\c\D_{B/A}$ and conclude that $\D_{\O/A}\in \I_B$.\\
Then prove $D_{\O/A}=N_{B/A}(\c)D_{B/A}$ and $D_{\O/A}=N_{B/A}(\D_{O/A})$ (in either order)..

\item[\bf (f)] Prove that for $\O :=A[\alpha]$ we have $D_{\O/A}=(\disc(f))$, and that $B=A[\alpha]$ if and only if $D_{B/A}=D_{\O/A}$.

\item[\bf (g)] Let $\q$ be a prime of $B$ lying above a prime $\p$ of $A$ and suppose the corresponding residue field extension is separable.  Prove that
\[
e_\q-1 \le v_\q(\D_{B/A})\le e_\q-1 + v_\q(e_\q),
\]
and that the lower bound is an equality only when it coincides with the upper bound (in which case $B/A$ is tamely ramified at $\q$).

\item[{\bf (h)}] Show that the upper bound in (f) is essentially the best possible by exhibiting a wildly ramified degree-$p$ extension of $\Q_p$ for which the upper bound is achieved, and showing that in the family of wildly ramified degree-$p$ extensions of $\Fp((t))$ obtained by adjoining a root of $x^p+t^nx+t$ the valuation of the different ideal is unbounded as $n$ increases (note that in this case $v_\q(e_\q)=v_\q(p)=v_\q(0)=\infty$, since we are in characteristic $p$, so (f) holds but imposes no upper bound).

\item[{\bf (i)}] Let $p$ and $q$ be distinct primes congruent to $1\bmod 4$, let $K:=\Q(\sqrt{pq})$, and let $L:=\Q(\sqrt{p},\sqrt{q})$.  Prove that $\D_{L/K}$ is the unit ideal (so $L/K$ is unramified).
\end{itemize}

\subsection*{Problem 2. Student's choice (33 points)}

Solve one of the 33 point problems on Problem Set 6 (any of problems 3,4,5,6) that you did not solve (be sure to indicate which problem you are solving).

\subsection*{Problem 3. 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
\end{tabular}
\end{center}
\noindent
Please feel free to record any additional comments you have on the problem sets and the lectures, in particular, ways in which they might be improved.
\medskip

\noindent
\textbf{Collaborators/Sources}:

\begin{thebibliography}{99}
\bibitem{AW45}
Emil Artin and George Whaples, \href{http://www.ams.org/journals/bull/1945-51-07/S0002-9904-1945-08383-9/S0002-9904-1945-08383-9.pdf}{\textit{Axiomatic characterization of fields by the product formula for valuations}}, Bull. Amer. Math. Soc. \textbf{51} (1945), 469--492.

\bibitem{hahn}
Hans Hahn, \emph{\"Uber die nichtarchimedischen Gr\"ossensysteme}, Sitzungsberichte der K. Akademie der Wissenschaften, Vienna \textbf{116} (1907), 601--655.

\bibitem{nashwilliams}
Crispin St. J. A. Nash-Williams, \href{https://doi.org/10.1017/S0305004100003844}{\textit{On well-quasi-ordering finite trees}}, Proc. Cambridge Philos. Soc. \textbf{59} (1963), 833--835.

\bibitem{neumann}
Bernhard H. Neumann, \href{https://doi.org/10.1090/S0002-9947-1949-0032593-5}{\textit{On ordered division rings}}, Trans. Amer. Math. Soc. \textbf{66} (1949), 202--252.

\bibitem{passman}
Donald S. Passman, \href{https://mathscinet.ams.org/mathscinet-getitem?mr=470211}{\textit{The algebraic structure of group rings}}, Wiley, 1977.
\end{thebibliography}
\end{document}
