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\textbf{18.785 Number Theory\hfill Fall~2025}\\\vspace{4pt}
\textbf{Problem Set \#11\due{12/5/2025}}\\\vspace{-6pt}
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\subsection*{Description}


These problems are related to Lectures 18--20.
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 any combination of problems that sum to 100 points, or \textbf{alternatively, solve any combination of problems that sum to 200 points to receive credit for two problem sets}.  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.



\subsection*{Problem 1. Higher ramification groups (49 points)}
Let $A$ be a complete DVR with finite residue field; its fraction field $K$ is a nonarchimedean local field (Prop. 9.6).
Let $L$ be a finite Galois extension of $K$, let $G:=\Gal(L/K)$, and let~$B$ be the integral closure of $A$ in $L$, with maximal ideal $\q=(\pi)$.
Fix $\alpha\in B$ so that $B=A[\alpha]$ (via Theorem 10.14), and let $f\in A[x]$ be the minimal polynomial of $\alpha$.

The decomposition group $D_\q$ is equal to $G$ (since $\sigma(\q)=\q$ for all $\sigma\in G$), and the inertia subgroup is $I_\q:=\{\sigma\in G:\sigma(x)\equiv x\bmod \q\text{ for all }x\in L\}$ with order equal to the ramification index $e:=e_\q$.  For any integer $i\ge -1$ define
\[
G_i:= \{\sigma\in G:\sigma(x)\equiv x\bmod \q^{i+1} \text{ for all }x\in B\},
\]
so that $G_{-1}=G$ and $G_0$ is the inertia subgroup.
The group $G_i$ is the $i$th \emph{ramification group} of $G$ (in the lower numbering).  Define $i_G:G\to \Z \cup \{\infty\}$ by $i_G(\sigma):=v_\q(\sigma(\alpha)-\alpha)$.

\begin{enumerate}[{\bf(a)}]
\item Prove that $G_i=\{\sigma\in G:i_G(\sigma)\ge i+1\}$, show that $G_{i+1}$ is a normal subgroup of~$G_i$, and show that the groups $G_i$ are trivial for all sufficiently large $i$.
\end{enumerate}

\noindent
Recall that the different ideal $\mathcal D:=\mathcal D_{B/A}$ is equal to  $(f'(\alpha))$ and satisfies the bounds
\[
e-1\le v_\q(\mathcal D) \le e-1+v_\q(e),
\]
with $e-1=v_\q(\mathcal D)$ if and only if $v_\q(e)=0$, by Proposition 12.23 and Theorem 12.26.

\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{1}
\item Prove Hilbert's \emph{different formula}:
\[
v_\q(\mathcal D) = \sum_{\sigma\ne 1}i_G(\sigma) = \sum_{i\ge 0} (\#G_i-1).
\]
\end{enumerate}

\noindent
Let $U_0:= B^\times$ be the unit group of $B$, and for $i > 0$ define
\[
U_i:= 1+\q^i = \{x\in U_0:x\equiv 1\bmod \q^i\}.
\]
\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{2}
\item Show that $U_0/U_1\simeq (B/\q)^\times$ and that for $i>0$ we have $U_i/U_{i+1}\simeq \q^i/\q^{i+1}$ isomorphic to the additive group of $B/\q$.  Conclude that $L/K$ is tamely ramified if and only if $G_1$ is trivial and totally wildly ramified if and only if $G=G_1$.\footnote{The group $G_1$ is sometimes called the \emph{wild inertia group}.}

\item Show that for $i\ge 0$ the map $\phi\colon \sigma\mapsto \sigma(\pi)/\pi$ defines a homomorphism $\phi_i\colon G_i\to U_i$ that induces an injection $G_i/G_{i+1}\hookrightarrow U_i/U_{i+1}$ (in other words, the composition of $\phi_i$ with the quotient map $U_i\to U_i/U_{i+1}$ has kernel $G_{i+1}$).
Conclude that (i) $G_0/G_1$ is cyclic of order prime to $p$, (ii) $G_1$ is the unique $p$-Sylow subgroup of $G_0$, (iii) $G_i/G_{i+1}$ an abelian $p$-group for all $i\ge 1$, and (iv) $G=\Gal(L/K)$ is solvable.

\item Suppose that $\sigma\in G_i-G_{i+1}$ and $\tau\in G_j-G_{j+1}$ with $1\le i\le j$.  Show that $\phi(\sigma\tau)-\phi(\tau\sigma)\equiv (j-i)u\pi^{i+j}\bmod \q^{i+j+1}$ for some $u\in U_0$ and that this implies $\phi(\sigma\tau\sigma^{-1}\tau^{-1})\equiv 1+(j-i)u\pi^{i+j}\bmod \q^{i+j+1}$.  Then use this to show $i\equiv j\bmod p$.

\end{enumerate}

\noindent
Let $K=\Q_p$ with $p$ odd, and let $L/\Q_p$ be a totally wildly ramified abelian extension, so in the notation above, $\Gal(L/K)=G=G_0=G_1$, and $\mathcal D$ is the different ideal.

\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{5}
\item Show that $v_\q(\mathcal D)=2p-2$ if $[L:K]=p$ and $v_\q(\mathcal D)=3p^2-p-2$ if $[L:K]=p^2$.
\item Prove that $G$ is cyclic (hint: reduce to $[L:K]=p^2$ then show that if $H\le G$ has order $p$ then $H=G_{p+1}$ by computing the different of $L/L^H$ using \textbf{(b)} and \textbf{(f)}).
\end{enumerate}

\subsection*{Problem 2. Polar density (49 points)}

Let $K$ be a number field and let $S$ be a set of primes of $K$.
Recall the \emph{Dirichlet density}
\[
d(S):=\lim_{s\to 1^+}\frac{\sum_{\p\in S}\N(\p)^{-s}}{\sum_\p \N(\p)^{-s}} = \lim_{s\to 1^+}\frac{\sum_{\p\in S}\N(\p)^{-s}}{\log\tfrac{1}{s-1}},
\]
and the \emph{natural density}
\[
\delta(S):=\lim_{x\to\infty}\frac{\#\{\p\in S:\N(\p)\le x\}}{\#\{\p:\N(\p)\le x\}},
\]
which are defined whenever these limits exist.
As shown on Problem Set 9, if $\delta(S)$ exists then so does $d(S)=\delta(S)$ (you may use this fact even if you did not prove it).


\begin{definition}
The \emph{partial Dedekind zeta function} associated to $S$ is the complex function
\[
\zeta_{K,S}(s):=\prod_{\p\in S}(1-\N(\p)^{-s})^{-1}.
\]
If for some integer $n\ge 1$ the function $\zeta_{K,S}^n$ extends to a meromorphic function on a neighborhood of $1$, the \emph{polar density} of $S$ is defined by
\[
\rho(S):= \frac{m}{n},\qquad m=-\ord_{s=1}\zeta_{K,S}^n(s).
\]
\end{definition}

\begin{enumerate}[{\bf(a)}]
\item Show that $\rho(S)$ is well defined (so $m/n$ does not depend on the choice of $n$).
\item Prove that $\delta(S)=d(S)=\rho(S)$ whenever $\rho(S)$ exists.
\item Show that $\rho(S)=0$ when $S$ is finite and $\rho(S)=1$ when $S$ is cofinite.
\item Show that $S\subseteq T$ implies $\rho(S)\le \rho(T)$ whenever both densities exist.
\item Let $\mathcal P_1$ denote the set of degree-1 primes of $K$ (those of prime norm). Prove that $\rho(\mathcal P_1)=1$ and $\rho(S\cap\mathcal P_1)=\rho(S)$ whenever $S$ has a polar density.
\item Let $L/K$ be a Galois extension and let $\Spl(L/K)$ be the set of primes of $K$ that split completely in $L$. Prove that $\rho(\Spl(L/K))=1/[L:K]$.  Conclude that for $G=\Gal(L/K)$ and any normal subgroup $H\subseteq G$, the set $S$ of primes $\p$ of $K$ for which the Frobenius conjugacy class $\Frob_\p$ lies in $H$ has polar density $\rho(S)=\#H/\#G$.
\item Prove that if $L/K$ and $M/K$ are Galois extensions of $K$ then $\Spl(M)=\Spl(L)$ if and only if $M=L$.
\end{enumerate}


\subsection*{Problem 3. The Frobenius density theorem (49 points)}

Let $L/K$ be a Galois extension of number fields of finite degree $n$ with Galois group $G:=\Gal(L/K)$.
Recall that for each unramified prime $\p$ of $K$, the \emph{Frobenius class} $\Frob_\p$ is the conjugacy class of the Frobenius elements $\sigma_\q$ for $\q|\p$.

The \emph{Chebotarev density theorem} states that for any set $C\subseteq G$ stable under conjugation (a union of conjugacy classes), the set  of unramified primes $\p$ with $\Frob_\p\subseteq C$ has Dirichlet density $\#C/\#G$.\footnote{It also has this natural density, but this was proved later.}
In this problem you will prove the \emph{Frobenius density theorem}, which says essentially the same thing, but with a different notion of conjugacy.

\begin{definition}
Two elements $g$ and $h$ of a group $G$ are \emph{quasi-conjugate} if they generate conjugate subgroups $\langle g\rangle$ and $\langle h\rangle$.
\end{definition}

\begin{enumerate}[{\bf(a)}]
\item Show that quasi-conjugacy is an equivalence relation and that each quasi-conjugacy class in a group is a union of conjugacy classes.
\item Show that in the symmetric group $S_n$, each quasi-conjugacy class is actually a conjugacy class (so the Frobenius density theorem implies the Chebotarev density theorem in this case), but that this is generally not true for the alternating group~$A_n$.
\item Suppose $G$ is cyclic. For each $d|n$, let $S_d$ be the set of primes $\p$ of $K$ for which the primes $\q|\p$ have inertia degree $f_\q=d$.
Prove that the set $S_d$ has polar density $\rho(S_d)=\phi(d)/[L\!:\!K]$ and conclude that infinitely many primes of $K$ are inert in $L$.
\end{enumerate}
\noindent
Fix $\sigma\in G$, let $K'=L^\sigma$ be its fixed field, let $H=\langle\sigma\rangle\subseteq G$, and let $d=\#H$.
Recall that in any number field, a \emph{degree-1 prime} is a prime whose absolute norm is prime.
For each prime $\p$ of $K$ (resp. $K'$) that is unramified in $L$, let $\overline\Frob_\p$ denote the quasi-conjugacy class in~$G$ (resp. $H$) that contains the conjugacy class $\Frob_\p$.

\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{3}
\item Let $S'$ be the set of degree-1 primes $\p'$ of $K'$ for which $\p=\p'\cap \O_K$ is unramified in $L$ and for which
$\sigma\in \overline\Frob_{\p'}$.  Prove that $S'$ has polar density $\rho(S')=\phi(d)/d$.
\item Let $S$ be the set of unramified degree-1 primes $\p$ of $K$ for which $\sigma\in \overline\Frob_\p$.  Show that map $\p'\mapsto \p'\cap \O_K$ defines a surjective map $\pi\colon S'\to S$.
\item Show that the fibers of $\pi$ all have cardinality $[K':K]/c$, where $c$ is the number of distinct conjugates of $H$ in $G$.
\item Show that $S$ has polar density
\[
\rho(S)=\frac{c\phi(d)}{[L\!:\!K]}.
\]
\item Prove that for any set $C\subseteq G$ stable under quasi-conjugation the set of unramified primes $\p$ of $K$ with $\overline\Frob_\p\subseteq C$ has polar density $\#C/\#G$.
\end{enumerate}


\subsection*{Problem 4. The Hilbert symbol (98 points)}

Let $K$ be a local field whose characteristic is not $2$.

\begin{definition}
The \emph{local Hilbert symbol} is the map $(\cdot,\cdot)\colon K^\times/K^{\times 2}\times K^\times/K^{\times 2}\to \{\pm 1\}$
\[
(a,b) := \begin{cases}
1 &\text{if $ax^2+by^2=1$ has a solution in $K$},\\
-1 &\text{otherwise.}
\end{cases}
\]
Here and throughout this problem $a,b\in K^\times $ are understood to represent elements of $K^\times/K^{\times 2}$ whenever the context requires it.
\end{definition}
\begin{enumerate}[{\bf(a)}]
\item Prove that the Hilbert symbol satisfies:
\begin{enumerate}[{\bf(i)}]
\item $(a,b)=(b,a)$ (symmetry);
\item $(a,bc)=(a,b)(a,c)$ and $(ab,c)=(a,c)(b,c)$ (bilinearity);
\item For any $a\in K^\times$, if $(a,b)=1$ for all $b\in K^\times$ then $a\in K^{\times 2}$ (nondegeneracy).
\item $(a,1-a)=1$ (for $a\ne 1$) and $(a,-a)=1$ (Steinberg relations).
\end{enumerate}
\item In part (a) where (if anywhere) did you use the fact that $K$ is a local field?  Determine which of (i)-(iv) hold for all fields whose characteristic is not 2, and for those that do not, give explicit counterexamples.
\item Prove that for $a\not\in K^{\times 2}$ we have $(a,b)=1$ if and only if $b\in \N_{K(\sqrt{a})/K}(K(\sqrt{a})^\times)$.
\item Let $L/K$ be an abelian extension. Let $r_{L/K}\colon K^\times/\N_{L/K}(L^\times)\to\Gal(L/K)$ be the isomorphism given by Artin reciprocity, and $\langle\cdot,\cdot\rangle := \Gal(\Kbar/K)\times K^\times/K^{\times 2}\to \{\pm 1\}$ the Kummer pairing $\langle \sigma,a\rangle := \sigma(\sqrt{a})/\sqrt{a}$ (which can be applied to $\sigma\in \Gal(L/K)$ whenever $L\subset\Kbar$ contains $\sqrt{a}$).
Prove that the Hilbert symbol satisfies
\[
(a,b) = \left \langle r_{K(\sqrt{b})/K}(a),b\right\rangle.
\]

\item For $a,b,c\in K^\times$ prove $ax^2+by^2=c$ has a solution if and only if $(-ab,c)=(a,b)$.
\item For $a,b\in K^\times$ define the quaternion algebra $H_{a,b}$ as the $K$-algebra $K(i,j)$ with $i^2=a$, $j^2=b$, $ij=-ji$.
Show that $(a,b)=1$ if and only if $H_{a,b}\simeq \mathrm{M}_2(K)$, the $2\times 2$ matrix algebra over $K$ (such quaternion algebras are said to \emph{split}). Then show that $H_{a,b}\simeq H_{a,c}$ if and only if $[b]=[c]$ in $K^\times/N_{K(\sqrt{a})/K}(K(\sqrt{a})^\times)$ and deduce that the isomorphism class of $H_{a,b}$ depends only on the Hilbert symbol $(a,b)$.
\item Show that for archimedean $K$ we have $(a,b)=-1$ if and only if $K\simeq \R$ and $a,b<0$.
\item Suppose that $K$ is nonarchimedean with residue field of odd cardinality $q$.  Let $\O$ be its valuation ring, $\pi$ a uniformizer for $\O$.  Define the residue symbol
\[
\left(\frac{a}{\pi}\right):=\begin{cases}
1 & \text{if }\bar a\in \F_q^{\times 2},\\
-1 & \text{if }\bar a\not\in \F_q^{\times 2},
\end{cases}
\]
where $\F_q:=\O/(\pi)$ is the residue field (which does not depend on the choice of $\pi$).
For $a,b\in K^\times$, let $a=u_a\pi^\alpha$, $b=u_b\pi^\beta$ with $u_a,u_b\in \O^\times$.  Prove the \emph{reciprocity law}:
\[
(a,b) = (-1)^{\alpha\beta(q-1)/2}\left(\frac{u_a}{\pi}\right)^{\beta}\left(\frac{u_b}{\pi}\right)^\alpha.
\]

\item Now let $K$ be a global field of characteristic not $2$, and for each place $v$ of $K$ let $(a,b)_v$ denote the Hilbert symbol of the completion of $K$ at $v$.
Prove the \emph{product formula}, which states that
\[
\prod_v(a,b)_v=1
\]
for all $a,b\in K^\times$ (and in particular, $(a,b)_v=1$ for all but finitely many places $v$).
\end{enumerate}

\noindent
Useful references for this problem if you get stuck are \cite[Ch.\,III]{serre73} and \cite[\S 5.6]{voight}.


\subsection*{Problem 5. Profinite groups (98 points)}

Recall that a topological space is \emph{totally disconnected} if every pair of distinct points can be separated by open neighborhoods that partition the space; totally disconnected spaces are obviously Hausdorff.

\begin{enumerate}[{\bf(a)}]
\item Show that products and inverse limits of totally disconnected compact topological spaces are totally disconnected and compact.  Conclude that every profinite group is a totally disconnected compact group.
\end{enumerate}

Let $G$ be a totally disconnected compact group, let $\widehat G:= \varprojlim G/N$ be its profinite completion (so $N$ varies over finite index open normal subgroups of $G$ ordered by containment), and let $\phi\colon G\to \widehat G$ be the natural map that sends each $g\in G$ to its images in the finite quotients $G/N$.

\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{1}
\item Show that every open subgroup of $G$ has finite index and contains an open normal subgroup (which necessarily also has finite index).
\item Show that $\phi(G)$ is both dense in $\widehat G$ and closed, hence equal to $\widehat G$; thus $\phi$ is surjective.
\item Show that to prove that $\phi$ is injective it suffices to show that the intersection of all open subgroups of $G$ is trivial.
Then show that for every $g\in G-\{1\}$ there is a neighborhood $U$ of $1$ that is both open and closed and does not contain $g$, and it is enough to show that every such $U$ contains an open subgroup $H$.
\item Let $U$ be a neighborhood of $1$ that is both open and closed.
Show that $U$ contains an open neighborhood of $1$ that is closed under multiplication and inversion, hence a subgroup (this requires some work; you will need to use the fact that the multiplication map $G\times G\to G$ is continuous and that $U$ is compact).
\item Show that $\phi$ is a continuous open map, hence a homeomorphism.  Conclude that $G$ is isomorphic to its profinite completion, and in particular, a profinite group.
\item Show that for a profinite group $G$ the following are equivalent: (i) the topology of $G$ is induced by a metric, (ii) $G\simeq \varprojlim G_n$, with $n\in \Z_{\ge 1}$, the $G_n$ finite, and $G_{n+1}\to G_n$ surjective, (iii) the number of open subgroups of $G$ is countable.
\item Show that the equivalent conditions (i)-(iii) in (g) imply that $G$ contains a countable dense subset (so $G$ is \emph{separable} as a topological space), and give an example showing that the converse does not hold.
\item Let $p$ be prime, let $\Z_p:=\varprojlim_n \Z/p^n\Z$, and let $G_p:=\prod_n \Z/p^n\Z$.
Show that $G_p$ and $\Z_p\times (G_p/\Z_p)$ are isomorphic as groups.  Are they isomorphic as topological groups?
\item Show that $\widehat \Z^\times \simeq \widehat\Z\times \prod_n\Z/n\Z$ as topological groups.
\end{enumerate}

\noindent
A useful reference for this problem if you get stuck is Ribes and Zalesskii~\cite{RZ}.

\subsection*{Problem 6. Arithmetically equivalent number fields (98 points)}

Two number fields are said to be \emph{arithmetically equivalent} if their Dedekind zeta functions coincide.
Isomorphic number fields obviously have the same zeta functions, but as proved by Gassmann \cite{gassmann}, the converse need not hold.
A particularly simple example is given by the fields $\Q(\sqrt[8]{a})$ and $\Q(\sqrt[8]{16a})$; as shown by Perlis \cite{perlis}, for $a\in \Z$ not square and not twice a square, these fields are arithmetically equivalent but nonisomorphic (they have the same Galois closure, which is obtained by adjoining a primitive 8th root of unity).

\begin{enumerate}[{\bf (a)}]
\item Show that arithmetically equivalent number fields must have the same Galois closure (up to isomorphism).
Conclude that if $K/\Q$ is Galois then $K'$ is arithmetically equivalent to $K$ if and only if $K'\simeq K$.
\end{enumerate}

In view of (a) we now consider two non-Galois extensions $K$ and $K'$ of $\Q$ with Galois closure $L/\Q$.
Let $G:=\Gal(L/\Q)$ and put $H:=\Gal(L/K)$ and $H':=\Gal(L/K')$.

\begin{enumerate}[{\bf (a)}]
\setcounter{enumi}{1}
\item Show that if $K$ and $K'$ are arithmetically equivalent number fields then for every prime $p$ there is a bijection between the primes of $K$ lying above $p$ and the primes of $K'$ lying above $p$ that preserves inertia degrees.
Conclude that $K$ and $K'$ must have the same degree.
\item Given a cyclic $C\subseteq G$, we can partition $G$ into double cosets $H\sigma_1C,\ldots,H\sigma_gC$.
When $H$ is not normal these cosets need not have the same size; each will have cardinality $f_i\cdot \#H$ for some integer $f_i\ge 1$.
Assume the $f_i$ are in increasing order and call the tuple $(f_1,\ldots,f_g)$ the \emph{coset type} of the pair $(H,C)$.  Prove that if $p$ is a prime that is unramified in $L$ and $C$ is any decomposition group of $p$ in $G$, then the coset type of $(H,C)$ is equal to the tuple of inertia degrees of the primes of $K$ lying above $p$ when arranged in increasing order.  Conclude that if $K$ and $K'$ are arithmetically equivalent then $(H,C)$ and $(H',C)$ have the same coset type for every cyclic subgroup $C$ of $G$.
\item Two subgroups $H$ and $H'$ of a finite group $G$ are said to be \emph{Gassmann equivalent} if for conjugacy class $c$ of elements in $G$ the sets $c\cap H$ and $c\cap H'$ have the same cardinality.
Show that this holds if and only if for every cyclic group $C$ the coset types of $(H,C)$ and $(H',C)$ coincide.
\item Suppose $H$ and $H'$ are Gassmann equivalent.  Show that $K$ and $K'$ have the same number of real and complex places (hint: consider the ``decomposition group" of the prime $p=\infty$ in $G$).
\end{enumerate}

\noindent
Like the Riemann zeta function, the Dedekind zeta function has a meromorphic continuation to $\C$ that satisfies a functional equation.
Define $\Gamma_\R(s):=\pi^{-s/2}\Gamma(s/2)$ and $\Gamma_\C(s):=2(2\pi)^{-s}\Gamma(s)$, and let
\[
Z_K(s):=|D_K|^{s/2}\Gamma_\R(s)^{n_1}\Gamma_\C(s)^{n_2}\zeta_K(s),
\]
where $n_1$ and $n_2$ are the number of real and complex places of $K$, respectively.
Then
\[
Z_K(s)=Z_K(1-s)
\]
as meromorphic functions on $\C$ (you may assume this in what follows).

\begin{enumerate}[{\bf (a)}]
\setcounter{enumi}{5}
\item Let $f(s)=f_1(s)/f_2(s)$ be a ratio of Euler products over a finite set of primes that extends to a meromorphic function on $\C$ that satisfies a functional equation $f(s)=g(s)f(1-s)$ for some meromorphic function $g(s)$ whose zeros and poles do not coincide with any zero or pole of $f$.  Prove that $g(s)=1$, then use this and the functional equations for $\zeta_K(s)$ and $\zeta_{K'}(s)$ to prove that if $H$ and $H'$ are Gassmann equivalent then $K$ and $K'$ are arithmetically equivalent.
\item A \emph{Gassmann triple} (or \emph{Gassmann-Sunada triple}) is a triple $(G,X,Y)$ in which $G$ is a group that acts faithfully on sets $X$ and $Y$ such that every element of $G$ fixes the same number of elements in $X$ and $Y$ but $X$ and $Y$ are not isomorphic as $G$-sets.
Show that $K$ and~$K'$ are arithmetically equivalent but not isomorphic if and only if $(G,G/H,G/H')$ is a Gassmann triple, where $G/H$ and $G/H'$ denote sets of cosets (for consistency with part (c), use right cosets and put the $G$-action on the right).
\item Show that if $K$ and $K'$ are arithmetically equivalent then they have the same normal core (largest subfield that is a normal extension of $\Q$).
Use this to show that~$K$ and $K'$ contain the same roots of unity and conclude that the unit groups $\O_K^\times$ and $\O_{K'}^\times$ are isomorphic as abelian groups.
\item Show that if $K$ and $K'$ are arithmetically equivalent then $|\disc\O_K|=|\disc \O_{K'}|$ and conclude that $h_KR_K=h_{K'}R_{K'}$, where $h_K:=\#\Cl\O_K$ is the class number and $R_K$ is the regulator of $K$ (and similarly for $K'$).
\end{enumerate}

\noindent
It is natural to ask whether the class numbers and regulators must also match.
This is not the case; the arithmetically equivalent fields $K:=\Q(\sqrt[8]{-15})$ and $K':=\Q(\sqrt[8]{-240})$ have class numbers 8 and 4, respectively (you do not need to prove this).
\begin{enumerate}[{\bf(a)}]
\setcounter{enumi}{9}
\item You showed in (b) that in arithmetically equivalent fields the multisets of inertia degrees above any prime $p$ must match (including at ramified primes); it is natural to ask whether the same is true of the ramification indices.
Show that this is not the case by showing that the polynomials $x^8-97$ and $x^8-16\cdot 97$ (which you may assume define arithmetically equivalent fields $K$ and $K'$), do not have the same factorization pattern in $\Q_2[x]$.
Conclude that the different ideals $\D_{K/\Q}$ and $\D_{K'/\Q}$ do not necessarily have the same factorization pattern, even though the discriminant ideals $D_{K/\Q}:=\N_{K/\Q}(\D_{K/\Q})$ and $D_{K'/\Q}:=\N_{K'/\Q}(\D_{K'/\Q})$ do.
\end{enumerate}
\noindent
A useful reference for this problem if you get stuck is Perlis' paper \cite{perlis}.

\subsection*{Problem 7. Survey (2 points)}
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
Problem 6 & & & \\\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{gassmann}
F. Gassmann, \href{http://link.springer.com/article/10.1007/BF01283860}{\textit{Bemerkungen zu der vorstehenden Arbeit von Hurwitz}} (comments on \textit{\"Uber Beziehungen zwischen den Primidealen eines algebraischen K\"orpers und den Substitutionen seiner Gruppe}, by Hurwitz) Math. Z. \textbf{25} (1926), 655-665.
\bibitem{perlis}
R. Perlis, \href{http://www.sciencedirect.com/science/article/pii/0022314X77900701}{\textit{On the equation $\zeta_K(s)=\zeta_{K'(s)}$}}, J. Number Theory \textbf{9} (1977), 342--360.
\bibitem{RZ}
L. Ribes and P. Zalesskii, \href{http://link.springer.com/book/10.1007/978-3-642-01642-4}{\textit{Profinite groups}}, 2nd edition, Springer, 2010.
\bibitem{serre73}
J.-P. Serre, \href{https://link.springer.com/book/10.1007/978-1-4684-9884-4}{\textit{A course in arithmetic}}, Springer, 1973.
\bibitem{voight}
J. Voight, \href{https://link.springer.com/book/10.1007/978-3-030-56694-4}{\textit{Quaternion algebras}}, Springer, 2021.
\end{thebibliography}
\end{document}