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\noindent
{\bf Math 254B: Number theory}

\noindent
H.$\,$W. Lenstra, Jr. (879 Evans, office hours MF 4:15--5:15,
tel. 643-7857, e-mail hwl@math).

\noindent
Spring 1993, MWF 1--2, 85 Evans.
\bigskip\noindent
{\bf Exercise 1.}\enspace
Let $L$ be a field and let $\Aut L$ be its automorphism group. We view
$\Aut L$ as a subset of the set $L^L$ of {\sl all\/} maps $L\to L$. We give
$L$ the discrete topology, $L^L$ the product topology, and $\Aut L$ the
induced (or relative) topology. In other words, a base for the topology of
$\Aut L$ is formed by the sets
$U_{\sigma,E}=\{\tau\in\Aut L:\tau|E=\sigma|E\}$, where $\sigma$ ranges over
$\Aut L$ and $E$ over the finite subsets of~$L$.

Prove that this topology makes $\Aut L$ into a topological group, i.$\,$e.,
both the map $\Aut L\times\Aut L\to\Aut L$ sending $(\sigma,\tau)$ to
$\sigma\tau$ and the map $\Aut L\to\Aut L$ sending $\sigma$ to $\sigma^{-1}$
are continuous.
\bigskip\noindent
{\bf Exercise 2.}\enspace
Let $L$ be a field, $K$ a subfield of $L$, and $G$ a subgroup of $\Aut L$.
Prove that $K\subset L$ is a Galois extension with group $G$ if and only if
$G$ is {\sl compact\/} in the topology induced from $\Aut L$ (see
Exercise 1) and $K=\{x\in L:\sigma x=x$ for all $\sigma\in G\}$.
\bigskip\noindent
In Exercises 3--6 we denote by $\Zhat$ the projective limit of the rings
$\ZZ/n\ZZ$ (with $n$ ranging over the positive integers) with respect to the
natural maps $\ZZ/n\ZZ\to\ZZ/m\ZZ$ (for $m$ dividing $n$). Explicitly, we have
$$\Zhat=\{(a_n)_{n=1}^\infty\in\prod_{n=1}^\infty(\ZZ/n\ZZ):
a_m=(a_n\bmod m)\hbox{ for all }n,\,m\hbox{ with }m\hbox{ dividing }n\}.$$
This is a compact topological ring. Its additive group is a profinite abelian
group, which is likewise denoted by $\Zhat$.
\bigskip\noindent
{\bf Exercise 3.}\enspace
Let $k$ be a finite field. Prove that the absolute Galois group of $k$ is
isomorphic to~$\Zhat$.
\bigskip\noindent
{\bf Exercise 4.}\enspace
Let $\mu$ be the group of all roots of unity in some algebraic closure of the
field $\QQ$ of rational numbers.\par
(a)\enspace Prove that $\Zhat$ is isomorphic to the endomorphism ring of the
abelian group~$\mu$.\par
(b)\enspace Prove that the group $\Zhat^*$ of units of $\Zhat$, with the
topology induced from $\Zhat$, is a profinite group, and that it is isomorphic
to the automorphism group of~$\mu$.\par
(c)\enspace Prove that $\QQ\subset\QQ(\mu)$ is a Galois extension, and that
$\Gal(\QQ(\mu)/\QQ)\cong\Zhat^*$.
\bigskip\noindent
{\bf Exercise 5.}\enspace
Let $a$ be a non-negative integer, and define $(a_n)_{n=0}^\infty$ inductively
by $a_0=a$, $a_{n+1}=2^{a_n}$. Prove that the sequence $(a_n)_{n=0}^\infty$
has a limit in $\Zhat$, and that this limit is independent of~$a$.
\bigskip\noindent
{\bf Exercise 6}. (a)\enspace Let $n$ be a positive integer. Prove that
multiplication by $n$ induces an exact sequence
$0\to\Zhat\to\Zhat\to\ZZ/n\ZZ\to0$.\par
(b)\enspace Let $u\in\Zhat^*$. Prove that for each positive integer $n$
there exists a unique element $a_n\in\Zhat$ such that $u^{n!}=1+n!a_n$, and
that $a_{n+1}\equiv a_n\bmod n!/\gcd(n!,n+1)$ for all~$n$.\par
(c)\enspace Prove that there is a continuous group homomorphism
$\log\colon\Zhat^*\to\Zhat$ defined by
$$\log u=\lim_{n\to\infty}{u^{n!}-1\over n!}.$$
\bigskip\noindent
{\bf Exercise 7.}\enspace
Let $k$ be a field, and let $k(X)$ be the field of fractions of the polynomial
ring $k[X]$ in one variable over~$k$.\par
(a)\enspace Determine all valuations $\psi$ on $k(X)$ for which the
restriction of $\psi$ to $k$ is trivial.\par
(b)\enspace Let $\varphi$ be a non-archimedean valuation on $k$, and let
$r$ be a positive real number. Prove that there is a valuation $\psi$ on
$k(X)$ for which $\psi|k=\varphi$ and $\psi(X)=r$.
\bigskip\noindent
{\bf Exercise 8.}\enspace
In this exercise you may assume the following fact, which will be proved in
class: if $\varphi$ is a valuation on a field $K$, and $L$ is a finite
extension of $K$, then $\varphi$ can be extended to a valuation on~$L$.\par
Let $K$ be a field. Prove: there exists a non-trivial non-archimedean
valuation on $K$ if and only if there does not exist a prime number $p$ with
the property that $K$ is an algebraic extension of~$\FF_p$.
\bigskip\noindent
{\bf Exercise 9.} (a)\enspace
Let $K$ be a field, $\varphi$ a non-archimedean valuation on $K$, and $n$ a
positive integer. Denote by $S_h$ the set of those non-zero vectors
$(x_1,x_2,\ldots,x_n)\in K^n$ with the property that $h$ is the smallest of
the subscripts $i$ for which $\varphi(x_i)=\max\{\varphi(x_j):1\le j\le n\}$.
Prove that any sequence $v_1$, $v_2$, \dots, $v_n$ of vectors in $K^n$
satisfying $v_i\in S_i$ for each $i$ forms a basis for $K^n$ over~$K$.\par
(b)\enspace Prove that the two-dimensional Euclidean plane can be written as
the union of three dense subsets with the property that no line in the plane
intersects all three subsets.
\bigskip\noindent
{\bf Exercise 10.}\enspace
Denote by $\CC$ the field of complex numbers, and let $\varphi$ be the
valuation on $\CC$ given by $\varphi(x)=|x|$ for all $x\in\CC$. Let $K$ be
a field, and let $\sigma$, $\tau\colon K\to\CC$ be field homomorphisms. Prove
that $\varphi\cdot\sigma=\varphi\cdot\tau$ if and only if $\sigma=\tau$ or
$\sigma=\bar\tau$; here the overhead bar denotes complex conjugation.
\bigskip\noindent
%factorial number system + metric + notation + negative numbers.
%chinese r t for Zhat.
{\bf Exercise 11.} (a)\enspace
Prove that for any sequence of elements $(c_n)_{n=1}^\infty$ of elements of
$\Zhat$ the sum $\sum_{n=1}^\infty c_nn!$ converges, and that for any
$x\in\Zhat$ there exists a unique sequence $(c_n)_{n=1}^\infty$ of integers
for which
$$x=\sum_{n=1}^\infty c_nn!,\qquad 0\le c_n\le n\quad\hbox{for all }n.$$
In this case, we shall write $x=(\ldots c_3c_2c_1)_!$. This representation
of elements of $\Zhat$ is called the {\sl factorial number system}.\par
(b)\enspace Write $-1$ in the factorial number system.\par
(c)\enspace Prove that the topology on $\Zhat$ is induced by the metric
$$d((\ldots b_3b_2b_1)_!,(\ldots c_3c_2c_1)_!)=1/\min\{n:b_n\ne c_n\}$$
($=0$ if $b_n=c_n$ for all $n$).
\bigskip\noindent
{\bf Exercise 12}.\enspace
For a prime number $p$, let $\ZZ_p$ be the projective limit of the rings
$\ZZ/p^n\ZZ$ (with $n$ ranging over the positive integers) with respect to the
natural maps $\ZZ/p^n\ZZ\to\ZZ/p^m\ZZ$ (for $m\le n$). This is a compact
topological ring, the {\sl ring of $p$-adic integers}.\par
Prove that there is a topological ring isomorphism
$$\Zhat\cong\prod_p\ZZ_p,$$
with $p$ ranging over the set of prime numbers and the product being given
the product topology.
\bigskip\noindent
{\bf Exercise 13.}\enspace
Prove that the field $\QQ(\ii)$, with $\ii^2=-1$, has a valuation $\varphi$
with the following property: if $x$, $y\in\QQ(\ii)$ are non-zero elements with
$\varphi(x+y)=\max\{\varphi(x),\varphi(y)\}$, then $\varphi(x)\ne\varphi(y)$.
\bigskip\noindent
{\bf Exercise 14.}\enspace Let $K$ be a field.\par
(a)\enspace Let $\varphi$ be a non-archimedean valuation on $K$, with
valuation ring $A$ and residue class field~$k$. Let $\zeta\in K$ be a root of
unity. Prove that $\zeta\in A$, and that $\zeta$ belongs to the kernel of the
natural map $A^*\to k^*$ if and only if the order of $\zeta$ is a power of the
characteristic of~$k$.\par
(b)\enspace Suppose that $\varphi$, $\psi$ are non-archimedean valuations
on $K$ with finite residue class fields $k$, $l$, and that
$\gcd(\#k,\#l)=1$. Prove that the number of roots of unity in $K$ is
finite, and that it divides
$\gcd\bigl((\#k)^{\#l}(\#k-1),(\#l)^{\#k}(\#l-1)\bigr)$.
\bigskip\noindent
{\bf Exercise 15.}\enspace
Let $K$ be a field that is complete with respect to a non-archimedean
valuation $\varphi$, and let $n$ be an integer that is not divisible by the
characteristic of the residue class field of~$\varphi$. Prove that the map
sending $x$ to $x^n$ is an automorphism of the multiplicative subgroup
$\{x\in K:\varphi(x-1)<1\}$ of~$K^*$.
\bigskip\noindent
{\bf Exercise 16.}\enspace
Let $K$ be a field and let $\varphi$ be a non-archimedean valuation on~$K$.
Suppose that the characteristic $p$ of the residue class field of $\varphi$ is
non-zero.\par
(a)\enspace Let $x\in K$, $\varphi(x-1)<1$. Prove that
$\varphi(x^p-1)\le\max\{\varphi(p\cdot(x-1)),\varphi(x-1)^p\}$.\par
(b)\enspace Suppose that $K$ is complete with respect to $\varphi$, and let
$\epsilon\in\RR$, $0<\epsilon<\varphi(p)^{1/(p-1)}$. Prove that the
map sending $x$ to $x^p$ is an isomorphism from the group
$\{x\in K:\varphi(x-1)\le\epsilon\}$ to the group
$\{x\in K:\varphi(x-1)\le\varphi(p)\cdot\epsilon\}$.
\bigskip\noindent
In the following two exercises we let $p$ be a prime number, and we denote by
$\QQ_p$ the field of $p$-adic numbers. The {\sl $p$-adic expansion\/} of a
$p$-adic number $r$ is the unique expansion $r=\sum_{i\in\ZZ}c_ip^i$ in
$\QQ_p$ for which $c_i\in\{0,1,\ldots,p-1\}$ for all $i$ and $c_i=0$ for
$i\ll0$.
\bigskip\noindent
{\bf Exercise 17.}\enspace
How can you tell from the $p$-adic expansion of a $p$-adic number $r$ whether
$r$ belongs to~$\ZZ$? and to $\half+\ZZ$?
\bigskip\noindent
{\bf Exercise 18.}\enspace
Let $r$ be a positive rational number. Prove the correctness of the following
method for obtaining the $p$-adic expansion of $r$. How does it have to be
modified for negative~$r\in\QQ$?

Write $r$ in $\RR$ in base $p$, say $r=\sum_{i\in\ZZ}a_ip^i$, with
$a_i\in\{0,1,\ldots,p-1\}$, $a_i=0$ for $i\gg0$. It is well known that the
sequence $(a_i)_{i=-1}^{-\infty}$ is ultimately periodic, say with a period
consisting of $m$ digits. Let $b_i\in\{0,1,\ldots,p-1\}$, for $i\in\ZZ$, be
given by $b_i=a_{i-km}$ for $k\in\ZZ$ sufficiently large, so that
$(b_i)_{i\in\ZZ}$ is purely periodic. The $p$-adic expansion
$\sum_{i\in\ZZ}c_ip^i$ of $r$ is now obtained by subtracting the expression
$\sum_{i\in\ZZ}b_ip^i$ from $\sum_{i\in\ZZ}a_ip^i$ by means of the usual
subtraction method in base~$p$.

{\it Example.} Take $p=5$ and $r=42{8\over15}$. The representation of $r$ in
base $5$ in $\RR$ is given by
$$132.2313131313\ldots\;.$$
The purely periodic expression $\sum_{i\in\ZZ}b_ip^i$ can be written as
$$\ldots1313131313.1313131313\ldots\;.$$
Subtracting this expression from the previous one in base $5$, going from
right to left, we find
$$\ldots3131313314.1.$$
This is the $5$-adic expansion of $42{8\over15}$; so $c_i=0$ for $i<-1$,
$c_{-1}=1$, $c_0=4$, $c_1=1$, $c_2=3$, $c_i=3$ for odd $i\ge3$, and $c_i=1$
for even $i\ge4$.
\bigskip\noindent
{\bf Exercise 19.} (a)\enspace Let $K$ be a field with a non-archimedean
valuation~$\varphi$. Denote the valuation ring and its maximal ideal by $A$
and~$P$. Write $S$ for the set of those $x\in K$ for which $1+x$ has an
$n\/$th root in $K$ for infinitely many positive integers~$n$. Prove: if $K$
is complete with respect to $\varphi$ then $P\subset S$, and if $\varphi$ is
discrete than $S\subset A$.\par
(b)\enspace Let $K$ be a field that is complete with respect to a non-trivial
valuation~$\varphi$. Prove that any discrete valuation on $K$ is equivalent
to~$\varphi$.\par
(c)\enspace For $i=0$, $1$, let $K_i$ be a field that is complete with
respect to a discrete valuation. Prove that any field homomorphism
$K_0\to K_1$ of which the image is not contained in the valuation ring of
$K_1$ is continuous.
\bigskip\noindent
{\bf Exercise 20.}\enspace Let $\QQ_p$ be the completion of $\QQ$ with respect
to any (finite or infinite) prime divisor $p$ of~$\QQ$. Prove that $\QQ_p$ has
no field automorphism except the identity. Prove also that $\QQ_p$ and
$\QQ_{p'}$ are not isomorphic as fields for $p\ne p'$.
\bigskip\noindent
{\bf Exercise 21.}\enspace Let $L$ be a field that is complete with respect to
a discrete valuation $\psi$, and let $K$ be a subfield of $L$ for which
$K\subset L$ is finite and separable. Prove that $K$ is complete with respect
to the restriction of $\psi$ to~$K$.
\bigskip\noindent
{\bf Exercise 22.}\enspace
Let $A$ be a local ring with residue class field $k$, and let $g$,
$h\in A[X]$. Suppose that $g$ is monic, and that the images of $g$ and $h$ in
$k[X]$ are coprime. Prove that $gA[X]+hA[X]=A[X]$.
\bigskip\noindent
{\bf Exercise 23.}\enspace
Let $K$ be a field that is complete with respect to a non-archimedean
valuation. Denote the valuation ring and its maximal ideal by $A$ and $P$, and
suppose that the residue class field $A/P$ is a finite field of $q$ elements.
Prove that for every $a\in A$ there is a unique element $\omega(a)\in a+P$
satisfying $\omega(a)^q=\omega(a)$, and that one has
$\omega(a)=\lim_{n\to\infty}a^{q^n}$.
\bigskip\noindent
{\bf Exercise 24.}\enspace
Let $k$ be a field, and let $f\in k[X]$ be an irreducible separable
polynomial. Let $\varphi$ be a valuation on $k(X)$ such that the restriction
of $\varphi$ to $k$ is trivial and $\varphi(f)<1$. (This determines $\varphi$
up to equivalence; cf. Exercise 7(a).) Write $l=k[X]/fk[X]$. Prove that $f$
has a zero in the completion $k(X)_\varphi$, and that there is a topological
isomorphism of fields $k(X)_\varphi\cong l((Y))$ that is the identity on~$k$;
here $l((Y))$ denotes the field of formal Laurent series with coefficients
in~$l$.
\bigskip\noindent
In the following three exercises $K$ denotes a field with a non-archimedean
valuation $\varphi$, and $r$ is a positive real number.
\bigskip\noindent
{\bf Exercise 25.}\enspace For $f=\sum_ia_iX^i\in K[X]$, $f\ne0$, denote the
largest and the smallest value of $i$ for which
$\varphi(a_i)r^i=\max_j\varphi(a_j)r^j$ by $l_r(f)$ and $s_r(f)$,
respectively.\par
(a)\enspace Prove that $l_r$ and $s_r$ extend to group homomorphisms
$K(X)^*\to\ZZ$.\par
(b)\enspace Suppose that $K$ is algebraically closed, and let $f\in K[X]$,
$f\ne0$. Prove that the number of zeroes $\alpha$ of $f$ in $K$ with
$\varphi(\alpha)=r$, counted with multiplicities, is equal to $l_r(f)-s_r(f)$.
\bigskip\noindent
{\bf Exercise 26.}\enspace Let $f=\sum_ia_iX^i\in K[X]$, $f\ne0$. The
{\sl Newton polygon\/} of $f$ is defined to be the ``lower convex hull'' of
the points $(i,-\log\varphi(a_i))$, with $i$ ranging over all non-negative
integers for which $a_i\ne0$; more precisely, if $C\subset\RR\times\RR$ is the
convex hull of the set of those points, then the Newton polygon equals
$\{(x,y)\in C:$ there is no $(x,y')\in C$ with $y'<y\}$. The Newton polygon is
the union of finitely many line segments of different slopes.\par
(a)\enspace Draw, for each prime number $p$, the Newton polygon of
$3X^3-{6\over7}X^2+{3\over2}X+5\in\QQ[X]$ with respect to the $p$-adic
valuation of~$\QQ$.\par
(b)\enspace Prove: if $\log r$ occurs as the slope of one of the line segments
that constitute the Newton polygon of $f$, then $l_r(f)-s_r(f)$ (as defined in
the previous exercise) is equal to the length of the projection of that line
segment on the $x$-axis, and otherwise $l_r(f)-s_r(f)=0$.\par
{\it Remark.} Combining (b) with Exercise 25(b) one sees that the valuations
of the zeroes of $f$ (in some algebraic extension of $K$) can be read from
the Newton polygon of~$f$.
\bigskip\noindent
{\bf Exercise 27.}\enspace Let $f\in K[X]$, and suppose that $f(0)\ne0$.\par
(a)\enspace Suppose that $K$ is complete with respect to $\varphi$, and that
$f$ is irreducible. Prove that the Newton polygon of $f$ is a single line
segment.\par
(b)\enspace Suppose that the Newton polygon of $f$ intersects the set
$\ZZ\times(-\log\varphi(K^*))$ in exactly two points. Prove that $f$ is
irreducible.\par
(c)\enspace Prove that $3X^3-{6\over7}X^2+{3\over2}X+5$ is the product of two
irreducible factors in each of $\QQ_2[X]$ and $\QQ_7[X]$, that it is
irreducible in $\QQ_3[X]$, and that it is the product of three linear factors
in $\QQ_5[X]$. How does it factor in $\QQ[X]$?
\bigskip\noindent
In the following two exercises we denote by $\zeta_m$ a primitive $m$th root
of unity in some extension of the base field.
\bigskip\noindent
{\bf Exercise 28.}\enspace Let $K$ be a field that is complete with respect to
a discrete valuation, with a perfect residue class field~$k$.\par
(a)\enspace Let $m$ be a positive integer that is not divisible by $\char k$.
Prove that $K(\zeta_m)$ is unramified over~$K$.\par
(b)\enspace Suppose that $k$ is a finite field of $q$ elements. Prove that
every finite unramified extension $L$ of $K$ is of the form $K(\zeta_m)$, with
$m$ as in~(a), and that for any such $m$ the order of $(q\bmod m)$ in
$(\ZZ/m\ZZ)^*$ equals $[L:K]$.
\bigskip\noindent
{\bf Exercise 29.}\enspace Let $p$ be a prime number.\par
(a)\enspace Prove that for each positive integer $s$ the extension
$\QQ_p\subset\QQ_p(\zeta_{p^s})$ is totally ramified of
degree $(p-1)p^{s-1}$.\par
(b)\enspace Prove that $\QQ_p(\zeta_p)=\QQ_p((-p)^{1/(p-1)})$.
\bigskip\noindent
{\bf Exercise 30.}\enspace Let $K$ be a field that is complete with respect to
a discrete valuation, and suppose that the residue class field is perfect. Let
$K\subset L\subset M$ be finite field extensions.\par
(a)\enspace Denote by $E$ the maximal subextension of $K\subset M$ that is
unramified over~$K$. Prove that $E\cap L$ is the maximal subextension of
$K\subset L$ that is unramified over $K$, and that $E\cdot L$ is the maximal
subextension of $L\subset M$ that is unramified over~$L$.\par
(b)\enspace This is the same as (a), with ``unramified'' everywhere replaced
by ``tamely ramified''.
\bigskip\noindent
{\bf Exercise 31.}\enspace Let $K$ be a field that is complete with respect to
a discrete valuation with a finite residue class field. Prove that $K$ is
either isomorphic to a finite extension of $\QQ_p$, for some prime number $p$,
or to the field $k((t))$ of formal Laurent series over $k$, for some finite
field~$k$.
\bigskip\noindent
{\bf Exercise 32.}\enspace Let $K$ be a field that is complete with respect to
a discrete valuation with a perfect residue class field. Let $K\subset L$ be a
finite Galois extension, with Galois group, inertia group, and ramification
group $G$, $I$, and $V$, respectively. Let $H\subset G$ be a subgroup, and
$E=L^H$ the corresponding subfield.\par
(a)\enspace Prove that $I\cap H$ and $V\cap H$ are the inertia group and the
ramification group of $E\subset L$, respectively.\par
(b)\enspace Suppose that $E$ is Galois over $K$, with Galois group $\Gamma$
($\cong G/H$). Prove that the images of $I$ and $V$ under the natural map
$G\to\Gamma$ are the inertia group and the ramification group of $K\subset E$,
respectively.
\bigskip\noindent
{\bf Exercise 33.}\enspace Do the following for $p=2$, $3$, $5$, $7$, and
$31$. Determine the degree of a splitting field $M$ of $X^3-2$ over $\QQ_p$,
as well as the Galois group, the inertia group, and the ramification
group of $\QQ_p\subset M$. In addition, compute $e(L/K)$ and $f(L/K)$ for all
fields $K$, $L$ with $\QQ_p\subset K\subset L\subset M$.
\bigskip\noindent
{\bf Exercise 34.}\enspace Let $K$ be as in Exercise 31, and suppose that the
characteristic of the residue class field of $K$ is different from~$2$. Let
the field $L$ be obtained by adjoining the square roots of all elements of
$K$ to~$K$. Prove that $L$ is a tamely ramified finite Galois extension of
$K$, the Galois group being the non-cyclic group of order~$4$. Can you
describe the maximal unramified subextension of $K\subset L$?
\bigskip\noindent
{\bf Exercise 35.}\enspace Let $K$ be as in Exercise 32, and let $L_1$, $L_2$
be two totally ramified finite extensions of $K$ inside a common overfield.
Does it follow that $L_1\cdot L_2$ is totally ramified over~$K$? Give a proof
or a counterexample.
\bigskip\noindent
{\bf Exercise 36.}\enspace Let $L=\QQ_5(\rott4\of{50})$, and let $E$ be the
maximal unramified subextension of $\QQ_5\subset L$. Exhibit a prime element
$\pi_E$ of the valuation ring of $E$ such that $L=E(\sqrt{\pi_E})$. Can
$\pi_E$ be chosen in~$\QQ_5$? Prove the correctness of your answer.
\bigskip\noindent
{\bf Exercise 37.}\enspace Let $G$ be a profinite group.\par
(a)\enspace Prove that for every $g\in G$ there exists a unique continuous
group homomorphism $f\colon\Zhat\to G$ with $f(1)=g$.\par
(b)\enspace Suppose that $N\subset G$ is a closed normal subgroup such that
$G/N\cong\Zhat$ (as profinite groups). Prove that $G$ is, as a profinite
group, isomorphic to a semidirect product of $\Zhat$ by~$N$.\par
(c)\enspace This is the same as (b), with $\Zhat$ replaced by~$\ZZ_p$ for
some prime number~$p$.
\bigskip\noindent
In the following three exercises $K$ denotes a field that is complete with
respect to a discrete valuation with a finite residue class field, and $L$ is
a finite extension of~$K$. The ramification index of $L$ over $K$ is denoted
by~$e$.
\bigskip\noindent
{\bf Exercise 38.}\enspace Let $M$ be the largest subextension of $K\subset L$
for which $K\subset M$ is abelian.\par
(a)\enspace Use local class field theory to show that $N_{L/K}L^*=N_{M/K}M^*$.
\par
(b)\enspace Let $q$ denote the cardinality of the residue class field of~$K$.
Suppose that $K\subset L$ is totally ramified, and that $\gcd(e,q(q-1))=1$.
Prove that $M=K$ and that $N\colon L^*\to K^*$ is surjective. Can you prove
both statements without local class field theory?
\bigskip\noindent
{\bf Exercise 39.} (a)\enspace Suppose that $\char K=0$. Prove that every
subgroup of $K^*$ of finite index is open. ({\it Hint\/}: Exercises 15 and
16.)\par
(b)\enspace Suppose that $\char K\ne0$. Prove that $K^*$ has a subgroup of
finite index that is not open.\par
(c)\enspace Let $H$ be a subgroup of $\RR^*$ or $\CC^*$. Prove that $H$ is
of finite index if and only if $H$ is open.
\bigskip\noindent
{\bf Exercise 40.} (a)\enspace Suppose that $L$ is unramified over~$K$.
Prove that $K^*/N_{L/K}L^*\cong\ZZ/[L:K]\ZZ$. (Do not use local class field
theory.)\par
(b)\enspace Suppose that $L$ is abelian over $K$, and denote the rings of
integers of $K$ and $L$ by $A$ and $B$, respectively. Use local class field
theory to show that $A^*/N_{L/K}B^*$ is of order $e$.
\bigskip\noindent
{\bf Exercise 41.}\enspace Do the following for $p=2$, $3$, $5$, $7$, and $31$.
Let $M$ be a splitting field of $X^3-2$ over $\QQ_p$ (cf. Exercise 33).
Describe $N_{M/\QQ_p}M^*$ as a subgroup of~$\QQ_p^*$. (You may use local class
field theory.)
\bigskip\noindent
{\bf Exercise 42.}\enspace Let $K$ be a field that is complete with respect to
a discrete valuation with a perfect residue class field, and let $L$ be a
finite separable extension of $K$. Denote the ramification index of $L$
over $K$ by $e$, the maximal ideal of the valuation ring of $L$ by $Q$, and
the different of $L$ over $K$ by $\Dd_{L/K}$.\par
(a)\enspace Prove: $eQ^{e-1}\subset\Dd_{L/K}$.\par
(b)\enspace Suppose that $\char K=p\ne0$, and let $m(L/K)\in\ZZ$ be such
that $\Dd_{L/K}=Q^{m(L/K)}$. Prove that $m(L/K)\not\equiv-1\bmod p$.\par
\bigskip\noindent
{\bf Exercise 43.}\enspace Let $p$ be a prime number, and let $\FF_p((t))$ be
the field of formal Laurent series over the field $\FF_p$ of $p$ elements.\par
(a)\enspace Prove that there exists an element $u\in\FF_p((t))$ that is
transcendental over the subfield $\FF_p(t)$ generated by~$t$.\par
(b)\enspace Let $u$ be as in (a), and put $K=\FF_p(t,u^p)$, $L=\FF_p(t,u)$.
Prove that $[L:K]=p$, and that $K$ has a discrete valuation $\varphi$ with
the property that $\sum_\psi e(\psi/\varphi)f(\psi/\varphi)<[L:K]$, the sum
ranging over the valuations $\psi$ of $L$ that extend~$\varphi$.
\bigskip\noindent
In Exercises 44--48 we let $K$ be an algebraic number field, i.$\,$e., a
finite extension of~$\QQ$. We let $K\subset L$ be a finite Galois extension
and $G$ its Galois group. By $Q$ we denote a prime of $L$, by $P$ its
restriction to $K$, by $G_Q$ the decomposition group of $Q$ over $P$, and by
$I_Q\subset G_Q$ the inertia group of $Q$ over~$P$. Let $H\subset G$ be a
subgroup, $E=L^H$ the corresponding subfield of $L$, and $R$ the restriction of
$Q$ to~$E$. We write $X$ for the transitive $G$-set $G/H$ belonging to~$E$;
it may alternatively be thought of as the set of $K$-homomorphisms
$E\to L$, or, if $E\cong_KK[t]/fK[t]$, as the set of zeroes of $f$ in~$L$.
\bigskip\noindent
{\bf Exercise 44.} (a)\enspace Prove that $G_Q\cap H$ is the decomposition
group of $Q$ over $R$.\par
(b)\enspace Suppose that $H$ is normal in $G$. Prove that the decomposition
group of $R$ over $P$ is the image of $G_Q$ in the group $G/H$.
\bigskip\noindent
{\bf Exercise 45.}\enspace Suppose that $Q$ is unramified over $P$, and
denote by $[L/K,Q]$ the Frobenius automorphism of $Q$ over~$P$.\par
(a)\enspace Prove that $[L/E,Q]=[L/K,Q]^{f(R/P)}$. \par
(b)\enspace Suppose that $H$ is normal in $G$. Prove that $[E/K,R]$ is the
restriction of $[L/K,Q]$ to~$E$.
\bigskip\noindent
{\bf Exercise 46.}\enspace Prove that there is a bijection between the set
of extensions $R'$ of $P$ to $E$ and the set of orbits $Y$ of $X$ under the
action of $G_Q$, with the following property. If $R'$ corresponds to $Y$, then
the length $\#Y$ of the orbit $Y$ equals $e(R'/P)f(R'/P)$, and $Y$ is the
disjoint union of $f(R'/P)$ orbits of length $e(R'/P)$ under the action
of~$I_Q$.
\bigskip\noindent
{\bf Exercise 47.}\enspace Suppose that $G$ is isomorphic to the symmetric
group $S_5$ of order $120$, that $G_Q$ has order $6$, and that $I_Q$ has
order $2$.\par
(a)\enspace Prove that, if the identification of $G$ with $S_5$ is suitably
chosen, $G_Q$ is generated by the permutation $(1\>2\>3)(4\>5)$ and $I_Q$ by
$(4\>5)$.\par
(b)\enspace Suppose that $[E:K]=5$. How many extensions $R'$ does $P$ have to
$E$, and what are the numbers $e(R'/P)$ and $f(R'/P)$?\par
(c)\enspace Suppose that $[E:K]=15$. How many extensions $R'$ does $P$ have to
$E$, and what are the numbers $e(R'/P)$ and $f(R'/P)$?
\bigskip\noindent
{\bf Exercise 48.}\enspace Suppose that $G$ is isomorphic to the symmetric
group $S_4$ of order~$24$, and that $Q$ is the {\sl only\/} prime of $L$
extending $P$.\par
(a)\enspace Prove that $P$ is $2$-adic, in the sense that the restriction of
$P$ to $\QQ$ is the $2$-adic prime of~$\QQ$.\par
(b)\enspace Determine $G_Q$ and $I_Q$ as subgroups of~$S_4$.\par
(c)\enspace Suppose that $H$ is cyclic of order~$4$. Determine $e(Q/R)$,
$f(Q/R)$, $e(R/P)$, and $f(R/P)$.
\bigskip\noindent
{\bf Exercise 49.}\enspace Let $m$ be a positive integer, and denote by
$\zeta$ a primitive $m$th root of unity in some extension of the base field.

(a)\enspace Let $K$ be an algebraic number field, let $P$
be a non-zero prime ideal of the ring of integers of $K$, and $\Nm P$ its norm
(i.$\,$e., the cardinality of the residue class field of~$P$). Prove that if
$m\notin P$, then $P$ is unramified in $K(\zeta)$, and the Artin symbol
$(K(\zeta)/K,\,P)$ maps $\zeta$ to $\zeta^{\Nm P}$.\par
(b)\enspace Let $p$ be a prime number, and let $m_0$ be the largest divisor of
$m$ that is not divisible by~$p$. Prove that, under the isomorphism
$\Gal(\QQ(\zeta)/\QQ)\cong(\ZZ/m\ZZ)^*$ that sends $\sigma$ to $(a\bmod m)$
if $\sigma\zeta=\zeta^a$, the inertia group of $p$ for the extension
$\QQ\subset\QQ(\zeta)$ corresponds to $\{a\in(\ZZ/m\ZZ)^*:a\equiv1\bmod m_0\}$,
and the decomposition group to $\{a\in(\ZZ/m\ZZ)^*:a\equiv p^i\bmod m_0$ for
some $i\in\ZZ\}$. What is the decomposition group of the infinite prime for
the same extension?
\bigskip\noindent
{\bf Exercise 50.}\enspace Let $\QQ\subset L$ be a finite Galois extension
with group $G$. Prove that $G$ is, as a group, generated by the inertia groups
$I_P$ of the non-archimedean primes $P$ of $L$ over $\QQ$.
\bigskip\noindent
{\bf Exercise 51.}\enspace Let $n$ be a positive integer, and denote by $S_n$
the symmetric group of degree~$n$.\par
(a)\enspace Let $G\subset S_n$ be a transitive subgroup that is generated by
transpositions. Prove that $G=S_n$.\par
(b)\enspace Let $K$ be an algebraic number field with $n=[K:\QQ]$. Suppose
that the discriminant of $K$ over $\QQ$ is squarefree. Prove that the Galois
closure $L$ of $K$ over $\QQ$ has degree $n!$, and that $\Gal(L/\QQ)\cong S_n$.
\bigskip\noindent
{\bf Exercise 52.}\enspace Let $K\subset L$ be a finite Galois extension of
algebraic number fields, with Galois group~$G$, and with rings of integers
$A\subset B$. For $\sigma\in G$, let $J_\sigma$ be the $B$-ideal generated
by $\{\sigma b-b:b\in B\}$. Let $H\subset G$ be a subgroup, and let $E=L^H$ be
the corresponding intermediate field. We write $X$ for the transitive $G$-set
$G/H$ corresponding to $E$.\par
(a)\enspace Prove: if $\Dd_{E/K}$ denotes the different of $E$ over $K$, then
$$\Dd_{E/K}B=\prod_{\sigma\in G-H}J_\sigma.$$
(You may assume the corresponding local fact.)\par
(b)\enspace Define $m(\sigma)=\#\{x\in X:\sigma x\ne x\}$ for $\sigma\in G$.
Prove: if $\Delta_{E/K}$ denotes the discriminant of $E$ over $K$, then
$$\Delta_{E/K}^{[L:K]}=
\prod_{\sigma\in G-\{1\}}(\Nm_{L/K}J_\sigma)^{m(\sigma)}.$$
\bigskip\noindent
{\bf Exercise 53.}\enspace Let $K$ be an algebraic number field. For an
irreducible monic polynomial $f\in K[t]$ we let $\delta(f)$ be the
discriminant of $K[t]/fK[t]$ over $K$; this is an ideal of the ring of
integers of~$K$. We extend the definition of $\delta(f)$ to all monic
polynomials $f$ by means of the rule $\delta(f_1)\delta(f_2)=\delta(f_1f_2)$.

(a)\enspace Let $f\in K[t]$ be a monic separable polynomial, and let $L$ be a
finite Galois extension of $K$ that contains a splitting field of~$f$. Express
$\delta(f)$ in terms of the ideals $J_\sigma$ from the previous exercise and
the numbers $\#\{\alpha\in L:f(\alpha)=0,$ $\sigma\alpha\ne\alpha\}$, for
$\sigma\in\Gal(L/K)$.\par
(b)\enspace Let $n$ be a positive integer and $a\in K^*$. Prove that
$\delta(t^n-a)$ is divisible by $\delta(t^n-1)$.
\bigskip\noindent
{\bf Exercise 54.}\enspace Let $K$ be an algebraic number field. Construct
a translation invariant metric on the adele ring $V_K$ of $K$ that induces
the given topology on $V_K$, and show that $V_K$ is complete with respect
to this metric.
\bigskip\noindent
In Exercises 55--59 we let $K$ be an algebraic number field and $A$ its
ring of integers. We write $\hat A$ for the projective limit of the finite
rings $A/nA$, with $n$ ranging over the set of positive integers. The idele
class group of $K$ is denoted by $C_K$, and the ideal class group of $A$
by $\Cl_A$.
\bigskip\noindent
{\bf Exercise 55.}\enspace Prove that $\hat A^*\cong\prod_{v<\infty}A_v^*$
(as topological groups), with $v$ ranging over the finite primes of $K$ and
$A_v$ denoting the valuation ring of the completion of $K$ at~$v$.
\bigskip\noindent
An exact sequence $0\to B\to C\to D\to0$ of topological abelian groups
with continuous group homomorphisms is said to {\sl split\/} if there
is an isomorphism $f\colon C\to B\times D$ of topological groups such
that (i) the map $B\to C\to B\times D$ is the canonical inclusion
$B\to B\times D$; and (ii) the map $C\to B\times D\to D$ is the given map
$C\to D$.
\bigskip\noindent
{\bf Exercise 56.}\enspace Denote by $V_\QQ$ the adele ring of~$\QQ$. In
class it was shown that there is an exact sequence
$0\to\Zhat\to V_\QQ/\QQ\to\RR/\ZZ\to0$ of topological abelian groups.\par
(a)\enspace Prove that $V_\QQ/\QQ$ is connected. Deduce that the sequence
does not split, even if in the definition given above the map $f$ is only
required to be an isomorphism of topological {\sl spaces\/} satisfying
(i) and~(ii).\par
(b)\enspace Prove that $V_\QQ/\QQ$ has a $\QQ$-vector space structure.
Deduce that the sequence does not split, even if in the definition given
above the map $f$ is only required to be a {\sl group\/} isomorphism
satisfying (i) and~(ii).
\bigskip\noindent
{\bf Exercise 57.}\enspace In class we proved that there is a surjective group
homomorphism $g\colon C_K\to\Cl_A$ such that the kernel $\ker g$ of $g$ fits
in an exact sequence
$$0\to\hat A^*\to\ker g\to(K\otimes_\QQ\RR)^*/A^*\to0\leqno(*)$$
of topological abelian groups.\par
(a)\enspace Suppose that $K\ne\QQ$. Prove that $A^*$ contains a unit $u\ne1$
that is {\sl totally positive\/} in the sense that $\sigma u>0$ for every
field homomorphism $\sigma\colon K\to\RR$. Prove also that for every such
$u$ there is a path in $\ker g$ from $1$ to $u$, where $u$ is now viewed as
an element of $\hat A^*$ via the natural inclusion $A^*\subset\hat A^*$.
(A {\sl path\/} is a continuous image of the unit interval $[0,1]$.)\par
(b)\enspace Prove that the sequence $(*)$ splits if and only if $K=\QQ$.
\bigskip\noindent
{\bf Exercise 58.}\enspace Let $D_K$ be the intersection of all open subgroups
of~$C_K$, and denote by $G_K^{\rm ab}$ the Galois group of the maximal
abelian extension of $K$ over~$K$.\par
(a)\enspace Prove that $C_K/D_K\cong G_K^{\rm ab}$.
(You may use class field theory.)\par
(b)\enspace Denote by $S$ the set of real places of $K$. Prove that there is
an exact sequence
$$0\to\bigl(\hat A^*\times\{\pm1\}^S\bigr)/\cl A^*\to
C_K/D_K\to\Cl_A\to0$$
of topological abelian groups; here $A^*$ is viewed as a subgroup of
$\hat A^*\times\{\pm1\}^S$ via the natural inclusion $A^*\subset\hat A^*$ and
the map $A^*\to\{\pm1\}^S$ that sends a unit to its signs under the real
embeddings of~$K$, and $\cl A^*$ denotes the closure of $A^*$ in
$\hat A^*\times\{\pm1\}^S$.\par
(c)\enspace What do (a) and (b) come down to for $K=\QQ$?
\bigskip\noindent
{\bf Exercise 59.}\enspace Let $K\subset E$ be a finite extension, and let $L$
be the largest subextension of $K\subset E$ for which $K\subset L$ is abelian.
Use the main theorem of class field theory to show that
$N_{E/K}C_E=N_{L/K}C_L$.
\bigskip\noindent
{\bf Exercise 60.}\enspace Let $k$ be a finite field, and let $K=k(t)$, where
$t$ is transcendental over~$k$. We write $A=k[t]$, and we let $\hat A$ be the
projective limit of the rings $A/fA$, with $f$ ranging over $A-\{0\}$. Let
$V_K$ and $J_K=V_K^*$ be the adele ring and the idele group of~$K$. We denote
by $k[[u]]$ the ring of power series in one variable $u$ over~$k$.\par
(a)\enspace Prove: $V_K/K\cong uk[[u]]\times\hat A$ as topological groups.
\par
(b)\enspace Prove: $J_K/K^*\cong\ZZ\times(1+uk[[u]])\times\hat A^*$ as
topological groups; here $1+uk[[u]]$ denotes the kernel of the map
$k[[u]]^*\to k^*$ that maps a power series to its constant term.
\bigskip\noindent
In Exercises 61--66 you may use the main theorem of class field theory. We let
$K$ be an algebraic number field. The class number of $K$ is denoted by~$h_K$.
\bigskip\noindent
{\bf Exercise 61.}\enspace Let $K\subset L$ be a finite abelian extension,
and let $v$ be a finite prime of~$K$. Prove that $v^2$ divides the conductor
of $L$ over $K$ if and only if $v$ is wildly ramified in~$L$.
\bigskip\noindent
{\bf Exercise 62.}\enspace Let $K\subset L$ be a finite abelian extension,
and let $v$ be a finite prime of $K$, lying over the rational prime~$p$.
Denote by $e$ the ramification index of $v$ over $p$, by $e'$ the least
integer $>e/(p-1)$, and by $k$ the number of factors $p$ in the exponent of
the inertia group of $v$ in $\Gal(L/K)$. Prove: if $v^m$ is the largest power
of $v$ dividing the conductor of $L$ over $K$, then $m\le e'+ke$.
({\it Hint\/}: use Exercise 16.)
\bigskip\noindent
{\bf Exercise 63.} (a)\enspace Prove that the Hilbert class field of
$\QQ(\sqrt{-15})$ is $\QQ(\sqrt{-3},\sqrt5)$.\par
(b)\enspace Prove that the Hilbert class field of $\QQ(\sqrt{-23})$ is the
splitting field of $X^3-X-1$ over~$\QQ$.
\bigskip\noindent
{\bf Exercise 64.}\enspace Let $E$, $F$ be two finite extensions of $\QQ$
inside a common overfield. Prove: if $h_E=h_F=1$, then $h_{E\cap F}=1$.
\bigskip\noindent
{\bf Exercise 65.}\enspace The {\sl Hilbert class field tower\/} of $K$ is the
sequence of fields
$K=H^{(0)}(K)\subset H^{(1)}(K)\subset H^{(2)}(K)\subset\ldots\subset
H^{(i)}(K)\subset\ldots\,$, where each $H^{(i)}(K)$ is the Hilbert class field
of $H^{(i-1)}(K)$. The Hilbert class field tower is said to be {\sl finite\/}
if $H^{(i+1)}(K)=H^{(i)}(K)$ for some~$i$. Prove that the Hilbert class field
tower of $K$ is finite if and only if there is a finite extension $E$ of $K$
with $h_E=1$.
\bigskip\noindent
{\bf Exercise 66.}\enspace Let $p$ be a prime number, and let $K\subset L$ be
an abelian extension of degree~$p$.\par
(a)\enspace Let $G$ be a $p$-group, i.$\,$e., a finite group of which the
cardinality is a power of $p$, and let $H$ be a subgroup of $G$ with
$H\ne G$. Prove that $G$ has a normal subgroup $N$ with $H\subset N$,
$N\ne G$.\par
(b)\enspace Suppose that at most one prime of $K$ is ramified in $L$.
Prove: if $p$ divides $h_L$, then $p$ divides~$h_K$.\par
(c)\enspace Suppose that at least one prime of $K$ is ramified in $L$.
Prove: if $p$ divides $h_K$, then $p$ divides~$h_L$.
\bigskip\noindent
{\bf Exercise 67.}\enspace Let $G$ be a cyclic group, and let
$0\to A\to B\to C\to0$ be a short exact sequence of $G$-modules. Prove that
there is a map $C^G\to A_G$ such that the sequence
$0\to A^G\to B^G\to C^G\to A_G\to B_G\to C_G\to0$ is exact.
In which way does your map depend on the choice of a generator for~$G$?
\bigskip\noindent
{\bf Exercise 68.}\enspace Let $G$ be a finite cyclic group, $n$ the order of
$G$, and $q$ a positive rational number. Prove that there exists a $G$-module
$A$ for which the Herbrand quotient $Q(G,A)$ is defined and equal to $q$ if
and only if $q\in\ZZ[1/n]^*$.
\bigskip\noindent
{\bf Exercise 69.}\enspace Let $K\subset L$ be a finite Galois extension of
algebraic number fields, with a cyclic Galois group~$G$. Let $J_L$ be the
group of ideles of~$L$. Prove that $H^1(G,J_L)=0$.
\bigskip\noindent
In Exercises 70--72 you may use the main theorem of class field theory.
\bigskip\noindent
{\bf Exercise 70.}\enspace Let $K$ be an algebraic number field, and let $L$
be a finite extension of $K$ inside some algebraic closure of~$K$. Prove that
the norm map of class groups $\Cl_L\to\Cl_K$ is trivial if and only if $L$
contains the Hilbert class field of~$K$.
\bigskip\noindent
{\bf Exercise 71.}\enspace In this exercise and the following we let
$K=\QQ(\sqrt{-3})$ and $L=K(2^{1/3})$. We write $\mu_3$ for the subgroup of
$K^*$ generated by the cube root of unity $(-1+\sqrt{-3})/2$. The unique
primes of $K$ lying over $2$ and $3$ are denoted by $2$ and $\tt$,
respectively.\par
(a)\enspace Prove that $K\subset L$ is cyclic of degree $3$, and that the map
$\epsilon\colon\Gal(L/K)\rightarrow\mu_3$ sending $\sigma$ to
$\sigma(2^{1/3})/2^{1/3}$ is a group isomorphism.\par
(b)\enspace Show that the conductor $\ff_{L/K}$ divides $2\tt^4$.\par
(c)\enspace Let $\pp$ be a finite prime of $K$ not dividing $2\tt$, and let
$\Nm\pp$ be the cardinality of its residue class field. Prove that
$\epsilon((\pp,L/K))$ is the unique element of $\mu_3$ that is congruent to
$2^{(\Nm\pp-1)/3}$ modulo $\pp$.
\bigskip\noindent
{\bf Exercise 72.} (a)\enspace Show that $L$ is the ray class field of $K$
modulo $6$ ($=2\cdot\tt^2$).\par
(b)\enspace Let $p$ be a prime number, and let $m$ be the number of distinct
cube roots of $(2\bmod p)$ in $\FF_p$. Prove the following:
\item{}$m=0$ if and only if $p\equiv1\bmod3$ and $p$ is not of the form
$a^2+27b^2$, with $a$, $b\in\ZZ$;
\item{}$m=1$ if and only if $p\not\equiv1\bmod3$;
\item{}$m\not=2$;
\item{}$m=3$ if and only if $p=a^2+27b^2$ for certain $a$, $b\in\ZZ$.
\bigskip\noindent
In Exercises 73--78 we denote by $K$ an algebraic number field, with
discriminant $\Delta_K$ and with ring of integers~$A$.
\bigskip\noindent
{\bf Exercise 73.}
(a)\enspace Let $a\in A$. Suppose that for all but finitely many prime ideals
$\pp$ of $A$ there exists $b_\pp\in A$ with $a\equiv b_\pp^2\bmod\pp$. Prove
that there exists $b\in A$ with $a=b^2$.\par
(b)\enspace Let $m\in\ZZ$, $m>0$. Suppose that for all but finitely many
primes $v$ of $K$ the completion $K_v$ contains a primitive $m$th root of
unity. Prove that $K$ contains a primitive $m$th root of unity.
\bigskip\noindent
{\bf Exercise 74.}\enspace Let $l\in\ZZ$, $l>[K:\QQ]$. Prove that the number
of roots of unity in $K$ is equal to the greatest common divisor of the
numbers $\Nm\pp-1$, where $\pp$ ranges over all prime ideals of $A$ for which
$A/\pp$ has characteristic greater than~$l$; here we write $\Nm\pp=\#A/\pp$.
\bigskip\noindent
In Exercises 75--78 you may use the main theorem of class field theory.
\bigskip\noindent
{\bf Exercise 75.}\enspace Suppose that $K$ is cubic, i.$\,$e., $[K:\QQ]=3$,
and that $\Delta_K$ is not a square.\par
(a)\enspace Prove that the Galois closure $M$ of $K$ over $\QQ$ contains a
unique quadratic subfield $L$, and that $\Delta_L$ divides~$\Delta_K$.\par
(b)\enspace Prove that $\Delta_L=\Delta_K$ if and only if $M$ is totally
unramified over $L$, and that in this case the class number of $L$ is
divisible by~$3$.
\bigskip\noindent
{\bf Exercise 76.}\enspace Suppose that $K$ is quartic, i.$\,$e., $[K:\QQ]=4$,
and that $K$ does not contain a quadratic subfield.\par
(a)\enspace Prove that the Galois closure $M$ of $K$ over $\QQ$ contains a
cubic subfield $L$, which is uniquely determined up to isomorphism, and that
$\Delta_L$ divides~$\Delta_K$.\par
(b)\enspace Prove that $\Delta_L=\Delta_K$ if and only if there is no finite
prime of the Galois closure of $L$ over $\QQ$ that ramifies in $M$, and that
in this case the strict class number of $L$ is even.
\bigskip\noindent
{\bf Exercise 77.}\enspace Let $d$ be a squarefree integer with
$d\equiv3\bmod8$, $d>3$. Prove that there exists a cubic field with
discriminant $-d$ or $-4d$.
\bigskip\noindent
{\bf Exercise 78.}\enspace Suppose that the class number of $K$ is a prime
number~$p$. Prove that the class number of the Hilbert class field of $K$ is
not divisible by~$p$.
\bigskip\noindent
In Exercises 79--84 we denote by $K$ an algebraic number field. You may
use the main theorem of class field theory.

Exercises 79 and 80 concern the following result, which was proved
in class as a consequence of the first inequality:

(A)\enspace{\sl if $K\subset L$ is a finite Galois extension, and $S$ is a
finite set of primes of $L$ containing the infinite ones and the primes that
ramify over $K$, then $\Gal(L/K)$ is generated by $\{[\qq,L/K]:\qq$ is a prime
of $L$, $\qq\notin S\}$}.

Actually a stronger theorem is valid:

(B)\enspace{\sl if $K\subset L$ is a finite Galois extension, then for every
element $\sigma\in\Gal(L/K)$ there exist infinitely many finite primes $\qq$
of $L$, unramified over $K$, such that $\sigma=[\qq,L/K]$.}

Statement (B) is a consequence of \v Cebotarev's density theorem (Nikolai
Gregorovi\v c \v Cebotarev, Russian mathematician, 1894--1947), and its only
known proof employs complex $L$-functions; see S. Lang, {\sl Algebraic number
theory}, Ch.\ VIII, \S 4, Theorem 10. (\v Cebotarev's density theorem itself
asserts a bit more, in that it makes ``infinitely many'' more precise.)
\bigskip\noindent
{\bf Exercise 79.} (a)\enspace Dirichlet's theorem on primes in arithmetic
progression asserts that for any two positive integers $a$, $m$ with
$\gcd(a,m)=1$ there exist infinitely many prime numbers $p$ with
$p\equiv a\bmod m$ (Peter Gustav Lejeune Dirichlet, German mathematician,
1805--1859). Show how to deduce this theorem from~(B).\par
(b)\enspace Show how to deduce the special case $m=24$, $a\not\equiv1\bmod24$
of Dirichlet's theorem from~(A).
\bigskip\noindent
{\bf Exercise 80.} (a)\enspace Let $G$ be a group acting transitively on a
finite set $\Omega$ with $\#\Omega>1$. Prove that there exists $\sigma\in G$
such that for all $\omega\in\Omega$ one has $\sigma\omega\not=\omega$.\par
(b)\enspace Let $f\in\ZZ[X]$ be an irreducible polynomial with the property
that $(f\bmod p)$ has a zero in $\FF_p$, for all but finitely many primes~$p$.
Use (B) to show that $f$ has degree~$1$.\par
(c)\enspace The classification of finite simple groups can be used to show
that, for {\sl finite} $G$, the element $\sigma$ in (a) can be chosen to be of
prime power order (see B. Fein, W. M. Kantor, M. Schacher, {\sl J. Reine
Angew.\ Math.\/} {\bf 328} (1981), 159--173). Prove that now (A) suffices to
deduce the statement in~(b).
\bigskip\noindent
{\bf Exercise 81.}\enspace Let $K\subset L$ be a finite extension for which
the group $K^*/N_{L/K}L^*$ is finite. Prove that $L=K$.
\bigskip\noindent
{\bf Exercise 82.}\enspace Let $\alpha$, $\beta$, $\gamma\in K$.\par
(a)\enspace Prove that there exist $x$, $y\in K$ with
$\alpha x^2+\beta y^2=\gamma$ if and only if for each prime $\pp$ of $K$ there
exist $x$, $y\in K_\pp$ with $\alpha x^2+\beta y^2=\gamma$.\par
(b)\enspace Suppose that $\alpha$, $\beta$ and $\gamma$ are non-zero. Prove
that for all but finitely many primes $\pp$ of $K$ there exist $x$,
$y\in K_\pp$ with $\alpha x^2+\beta y^2=\gamma$.
\bigskip\noindent
{\bf Exercise 83.}\enspace Let $n$ be a positive integer, and denote by
$\zeta_n$ a primitive $n$-th root of unity.\par
(a)\enspace Prove that the class number of $\QQ(\zeta_n+\zeta_n^{-1})$ divides
the class number of $\QQ(\zeta_n)$.\par
(b)\enspace Suppose that $n$ is a power of $2$, and that $K$ is a subfield of
$\QQ(\zeta_n)$. Prove that the strict class number of $K$ is odd.
\bigskip\noindent
{\bf Exercise 84.}\enspace Let $S$ be a set of primes of $K$ containing the
set of infinite primes, and let $L$ be the maximal abelian totally unramified
extension of $K$ in which all primes in $S$ split completely. Prove that there
is a subring $B\subset K$, which is a Dedekind ring with field of fractions
$K$, such that $\Gal(L/K)$ is isomorphic to the class group of~$B$.
\bye

