From hwl@math.berkeley.edu  Thu Apr 17 16:35:13 1997
Return-Path: <hwl@math.berkeley.edu>
Received: from math.berkeley.edu (math.Berkeley.EDU [128.32.183.94])
          by ferrari.autobahn.org (8.8.5/8.8.4) with ESMTP
	  id QAA19236 for <was@ferrari.autobahn.org>; Thu, 17 Apr 1997 16:35:13 -0700
Received: from contessa.berkeley.edu by math.berkeley.edu (8.7.5/1.33(math)Ow.3)
	id QAA23809; Thu, 17 Apr 1997 16:35:10 -0700 (PDT)
From: hwl@math.berkeley.edu (Hendrik Lenstra)
Received: (hwl@localhost) by contessa.berkeley.edu (8.7.5/8.6.4) id QAA25334 for was; Thu, 17 Apr 1997 16:35:09 -0700 (PDT)
Date: Thu, 17 Apr 1997 16:35:09 -0700 (PDT)
Message-Id: <199704172335.QAA25334@contessa.berkeley.edu>
To: was@math.berkeley.edu
Status: RO
X-Status: 

\vsize 10in
\magnification=\magstep1
\font\tengoth=eufm10
\font\sevengoth=eufm10 at 7pt
\font\fivegoth=eufm10 at 5pt
\newfam\gothfam \def\goth{\fam\gothfam\tengoth} \textfont\gothfam=\tengoth
\scriptfont\gothfam=\sevengoth \scriptscriptfont\gothfam=\fivegoth

\baselineskip=15pt
\lineskiplimit=3pt
\lineskip=3pt minus 1pt

\def\aa{{\goth a}}
\def\Cl{{\rm Cl}}
\def\dd{{\goth d}}
\def\ff{{\goth f}}
\def\Fp{{\bf F}_p}
\def\Ftwo{{\bf F}_2}
\def\Gal{{\rm Gal}}
\def\id{{\rm id}}
\def\legendre{\overwithdelims()}
\def\mm{{\goth m}}
\def\NN{{\goth N}}
\def\pp{{\goth p}}
\def\QQ{{\bf Q}}
\def\qq{{\goth q}}
\def\rk{{\rm rk}}
\def\ttt{{\goth t}}
\def\ZZ{{\bf Z}}

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

\noindent
H.W. Lenstra, Jr.\ (office 879, tel.\ 643.7857, office hours MF
3:15--4),

\noindent
Spring 1988, MWF 2-3, 6 Evans.
\vskip 12pt
\noindent
\noindent
This course is a continuation of {\bf Math 254A}
(Number theory), which was taught in the fall of 1987.
The contents of the latter course are supposed to be known.

The main subject of the course will be {\sl class field
theory}, which classifies the abelian extensions of
an algebraic number field.
Other subjects have not yet been decided upon, but they
will probably be taken from the following:
local class field theory;
power reciprocity laws;
quadratic fields;
$L$-series and distribution of prime ideals;
the Riemann hypothesis for curves over finite fields.

Every one or two weeks some homework problems will be handed
out, to accommodate students that wish to get credit
for the course. There will not be a final examination.
\vskip 12pt
\noindent
The main theorems of class field theory were obtained by Teiji Takagi
(Japanese mathematician, 1875-1960) and Emil Artin (German
mathematician, 1898-1962) in the period 1920--1926.
For the history of class field theory, see Chapter XI by H. Hasse
in the book by Cassels and Fr\"ohlich mentioned below.
The proofs of Takagi and Artin depended partly on analytic techniques
(Dirichlet $L$-series); see
H. Hasse,
{\sl Bericht \"uber neuere Untersuchungen und Probleme aus der Theorie
der algebraischen Zahlk\"orper}, I, Ia, II, {\sl Jber.\ dt.\ Mat.\ Verein.}
{\bf 35} (1926), 1--55; {\bf 36} (1927), 233--311; {\sl Erg.\ Bd.} {\bf 6}
(1930), 1--204; reprint: Physica-Verlag, W\"urzburg, 1965. 

A modern exposition of the treatment depending on $L$-series
is contained in
S. Lang, {\sl Algebraic number theory},
Addison-Wesley, Reading, 1970;
reprint: Graduate Texts in Mathematics {\bf 110}, Springer-Verlag,
New York, 1986.

After the war, an approach to the main theorems of class field theory
was found that does not depend on analysis but on {\sl cohomology
of groups}. The basic reference is
E. Artin, J. Tate,
{\sl Class field theory}, Benjamin, New York, 1967.
For a more accessible treatment, see
J.W.S. Cassels, A. Fr\"ohlich (eds),
{\sl Algebraic number theory},
Academic Press, London--New York, 1967;
paperback reprint, 1986.

In the course no particular textbook will be followed.
We shall adopt an approach that avoids both the
$L$-functions and the cohomological techniques. It goes back to
C. Chevalley, {\sl La th\'eorie des corps de classes, Ann.\ Math.}
{\bf 41} (1940), 394-417.

Other approaches to class field theory can be found in A. Weil,
{\sl Basic number theory}, Springer-Verlag, Berlin, 1967, and in
J. Neukirch, {\sl Class field theory}, Springer-Verlag, Berlin, 1986.
\vskip 12pt
\noindent
{\bf Exercise 1.}
In class it was shown how to deduce the quadratic reciprocity law
$${p\legendre q}=(-1)^{{p-1\over2}{q-1\over2}}{q\legendre p}$$
(for odd primes $p$, $q$) from the inclusion
$\QQ(\sqrt{{-1\legendre q}q})\subset\QQ(\zeta_q)$ and properties
of the Artin symbol. Give a similar proof for the supplementary law
$${2\legendre p}=\cases{1&if $p\equiv\pm1\bmod8$,\cr
                       -1&if $p\equiv\pm3\bmod8$,\cr}$$
where $p$ denotes an odd prime.
\vskip 12pt
\noindent
{\bf Exercise 2.}
Let $L$ be a field extension of $\QQ$ with $[L:\QQ]\leq2$, and
denote by $\Delta$ the absolute value of the discriminant of
$L$ over $\QQ$.
For a positive integer $m$, let $\zeta_m$ denote a primitive $m$-th
root of unity in an extension field of $\QQ$.

(a) Prove that $L$ is isomorphic to a subfield of $\QQ(\zeta_\Delta)$.

(b) (This is harder.) Prove that $\Delta$ is the minimum of all
positive integers $m$ for which
$L$ is isomorphic to a subfield of $\QQ(\zeta_m)$.
\vskip 12pt
\noindent
In Exercises 3--5 we let $f=X^3-X-1$, which is an
irreducible polynomial in $\ZZ[X]$ with discriminant $-23$.
We let $\alpha$ be a zero of $f$ in an algebraic closure of $\QQ$,
and we write $K=\QQ(\sqrt{-23})$, $L=K(\alpha)$.

In Exercises 4 and 5, the theorems stated (but not yet proved)
in class may be used.
\vskip 12pt
\noindent
{\bf Exercise 3.}
(a)\enspace Prove that $L$ is the Galois closure of
$\QQ(\alpha)$ over $\QQ$, and that $K\subset L$ is an
abelian extension of degree $3$.\par\noindent
(b)\enspace Prove that exactly two primes of $\QQ(\alpha)$
are ramified over $\QQ$, and that they lie over $23$ and $\infty$.
Prove that in both cases the ramification index equals $2$.
\par\noindent
(c)\enspace Prove that $K\subset L$ is totally unramified.
\vskip 12pt
\noindent
{\bf Exercise 4.}
Prove that $L$ is the Hilbert class field of $K$.
\vskip 12pt
\noindent
{\bf Exercise 5.}
Let $p$ be a prime number, and let $m$ be the number of
distinct zeros of $(f\bmod p)$ in $\Fp$.
Prove the following:\vskip 3pt\noindent
(a)\enspace$m=0$ if and only if ${p\legendre23}=1$ and $p$
cannot be written as $p=a^2+23b^2$ with $a$, $b\in\ZZ$.
\par\noindent
(b)\enspace$m=1$ if and only if ${p\legendre23}=-1$.
\par\noindent
(c)\enspace$m=2$ if and only if $p=23$.
\par\noindent
(d)\enspace$m=3$ if and only if $p$
can be written as $p=a^2+23b^2$ with $a$, $b\in\ZZ$, $a\not=0$.
\vskip 12pt
\noindent
In Exercises 6 and 7, we let $K$ be an algebraic number
field and $L$ a finite abelian extension of $K$.
Further we let $\pp$ be a prime of $K$.

Let $\ff(L/K)$ be the {\sl conductor\/} of $L$ over $K$;
so this is the greatest common divisor of
all cycles $\mm$ of $K$ for which $L$ is contained in the
ray class field modulo $\mm$ of $K$.
Denote by $m$ the exponent to which $\pp$ appears in $\ff$.
It is part of the main theorem of class field theory that
$m\geq1$ if and only if $\pp$ is ramified in $L$,
and that $m\geq2$ if and only if $\pp$ is 
finite and {\sl wildly\/} ramified in $L$.

In these exercises we shall find an {\sl upper\/} bound for $m$.
We may and do assume that $\pp$ is {\sl finite}.
Denote by $p$ be the prime of $\QQ$ over which $\pp$ is lying, and let
$e=e(\pp/p)$ be the ramification index of $\pp$ over $p$.

For exercise 7, the theorems stated (but not yet proved)
in class may be used.
\vskip 12pt
\noindent
{\bf Exercise 6.}
For an integer $i>0$, denote by $U_i$ the open subgroup
$1+\pp^i$ of $K_\pp^*$.
Prove the following assertions.

(a)\enspace If $i$, $j$ are positive integers with
$j\not\equiv0\bmod p$, then the map $U_i\rightarrow U_i$
sending every $x$ to $x^j$ is an isomorphism.

(b)\enspace If $i>e/(p-1)$, then there is an
isomorphism $U_i\rightarrow U_{i+e}$ sending every $x$ to $x^p$.

(c)\enspace If $j$ is a positive integer, then
$(K_\pp^*)^j$ is an {\sl open\/} subgroup of $K_\pp^*$, and it
contains $U_{e'+ke}$, where $e'$ denotes the least integer
$>e/(p-1)$ and $k$ is the number of factors $p$ in $j$.

(d)\enspace If $K_\pp\subset E$ is a finite extension,
then $N_{E/K_\pp}[E^*]$ is an open subgroup of $K_\pp^*$,
and it contains $U_{e'+ke}$, with $e'$ as in (c)
and $k$ the number of factors $p$ in $[E:K_\pp]$.
\vskip 12pt
\noindent
{\bf Exercise 7.}
(a)\enspace Prove that $m\leq e'+ke$, where $e'$ denotes the
least integer $>e/(p-1)$ and $k$ is the number of factors $p$
in $[L:K]$.

(b)\enspace More precisely, prove that $m\leq e'+ke$, with $e'$
as before, but with $k$ now equal to the number of factors $p$
in the exponent of the inertia group of $\pp$ in $\Gal(L/K)$.
\vskip 12pt
\noindent
Exercises 8--10 are devoted to the extension
$K\subset L$, where $K=\QQ(\sqrt{-3})$ and $L=K(2^{1/3})$.
We write $\zeta_3$ for the cube root of unity $(-1+\sqrt{-3})/2$
in $K$, and $\mu_3$ for the subgroup of $K^*$ generated by
$\zeta_3$.
The unique primes of $K$ lying over $2$ and $3$ are denoted by $2$ and
$\ttt$, respectively.

The theorems stated (but not yet proved) in class may be used.
\vskip 12pt
\noindent
{\bf Exercise 8.}
(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\noindent
(b)\enspace Show that the conductor $\ff(L/K)$ divides $2\ttt^4$.
\par\noindent
(c)\enspace Let $\pp$ be a finite prime of $K$ not dividing
$2\ttt$, and let $\NN\pp$ 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 $2^{(\NN\pp-1)/3}$
modulo $\pp$.
\vskip 12pt
\noindent
{\bf Exercise 9.}
Show that $L$ is the ray class field of $K$ with modulus
$6$ ($=2\cdot\ttt^2$).
\vskip 12pt
\noindent
{\bf Exercise 10.}
Let $p$ be a prime number, and let $m$ be the number of
distinct cube roots of $(2\bmod p)$ in $\Fp$.
Prove the following:\vskip 3pt\noindent
(a)\enspace$m=0$ if and only if $p\equiv1\bmod3$ and $p$
cannot be written as $p=a^2+27b^2$ with $a$, $b\in\ZZ$.
\par\noindent
(b)\enspace$m=1$ if and only if $p\not\equiv1\bmod3$.
\par\noindent
(c)\enspace$m\not=2$.
\par\noindent
(d)\enspace$m=3$ if and only if $p$
can be written as $p=a^2+27b^2$ with $a$, $b\in\ZZ$.
\vskip 12pt
\noindent
{\bf Exercise 11.}
Let $G$ be a finite cyclic group, $m$ the order of $G$, and
$A$ a $G$-module.\vskip 3pt\noindent
(a)\enspace Prove that $m$ annihilates $H^i(G,A)$ for $i=0$, $1$.
\par\noindent
(b)\enspace Suppose that for each $a\in A$ there is a unique 
$b\in A$ with $ma=b$. Prove that $H^i(G,A)=0$ for $i=0$, $1$.
\par\noindent
(c)\enspace Suppose that the Herbrand quotient $h(G,A)$ is defined.
Prove that every prime number dividing
the numerator or denominator of $h(G,A)$ divides
$m$. Can you prove that conversely every positive rational
number with this property occurs as $h(G,A)$, for suitable $A$?
\vskip 12pt
\noindent
{\bf Exercise 12.}
Let $K\subset L$ be a finite Galois extension of algebraic number
fields, and suppose that the Galois group $G$ is cyclic.
Let $J_L$ be the group of id\`eles of $L$.
Prove that $H^1(G,J_L)=\{1\}$.
\vskip 12pt
\noindent
{\bf Exercise 13.}
Let $K\subset L$ be a finite Galois extension of algebraic number
fields, with Galois group $G$. Denote by $C_K$ and $C_L$ the id\`ele
class groups of $K$ and $L$. Prove that $C_K=C_L^G$.
[{\sl Hint}. In class this was shown for {\sl cyclic} $G$.
Either generalize this proof, or reduce the general case to the
cyclic case.]
\vskip 12pt
\noindent
In Exercises 14--16, let $K$ be an algebraic number field,
$\Cl_K$ its ideal class
group, and $h_K=\#\Cl_K$ its class number.
We let $\bar K$ be an algebraic closure of $K$,
and $H(K)$ the {\sl Hilbert class field\/} of $K$,
which is the maximal abelian totally unramified extension of $K$
inside $\bar K$.

In class it was shown, as a consequence of the main theorem
of class field theory, that $H(K)$ is a finite abelian
extension of $K$ of degree $h_K$, and that the Artin symbol
induces a group isomorphism $\Cl_K\rightarrow\Gal(H(K)/K)$.

The exercises illustrate how properties of the Hilbert
class field can be used to obtain information about the
class number.

The theorems stated (but not yet proved) in class may be used.
\vskip 12pt
\noindent
{\bf Exercise 14.}
(a)\enspace
Let $E$ be a finite extension of $K$.
Prove that $H(K)\subset H(E)$, and that
$h_K$ divides $h_E\cdot[E:K]$.
\par\noindent
(b)\enspace 
Let $E$, $F$ be two finite extensions of $\QQ$ inside $\bar \QQ$.
Prove: if $h_E=h_F=1$, then $h_{E\cap F}=1$.
\vskip 12pt
\noindent
{\bf Exercise 15.}
(a)\enspace 
Let $E$ be a finite extension of $K$, and denote by $L$ the
maximal abelian totally unramified extension of $K$ inside $E$.
Show that the cokernel of the norm map
$N_{E/K}\colon\Cl_E\rightarrow\Cl_K\,$ is isomorphic
to $\Gal(L/K)$.
\par\noindent
(b)\enspace Let $n$ be a positive integer, and denote by
$\zeta_n$ a primitive $n$-th root of unity.
Prove that the class number of $\QQ(\zeta_n+\zeta_n^{-1})$
divides the class number of $\QQ(\zeta_n)$.
\vskip 12pt
\noindent
{\bf Exercise 16.}
The {\sl Hilbert class field tower\/} of $K$ is the sequence
of fields $H^{(0)}(K)=K\subset H^{(1)}(K)=H(K)\subset
H^{(2)}(K)=H(H(K))\subset \ldots\subset H^{(i)}(K)\subset\ldots\,$,
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 $h_E=1$.
\vskip 12pt
\noindent
Exercises 17--19 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 of
algebraic number fields, 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 let the hypotheses be the
same; 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.)
\vskip 12pt
\noindent
{\bf Exercise 17.}
(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\noindent
(b)\enspace 
Show how to deduce the special case $m=24$, $a\not\equiv1\bmod24$
of Dirichlet's theorem from (A).
\vskip 12pt
\noindent
{\bf Exercise 18.}
Let $f\in\ZZ[X]$ be a non-zero polynomial with the property
that the polynomial $(f\bmod p)\in\ \Fp[X]$ splits completely
into linear factors in $\Fp[X]$,
for all but finitely many prime numbers $p$.
Prove that $f$ splits completely into linear factors in $\QQ[X]$.
(Don't use (B).)
\vskip 12pt
\noindent
{\bf Exercise 19.}
(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\noindent
(b)\enspace 
Let $f\in\ZZ[X]$ be an irreducible polynomial with the property that
$(f\bmod p)$ has a zero in $\Fp$, for all but finitely many primes
$p$. Use (B) to show that $f$ has degree $1$.
\par\noindent
(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).
\vskip 12pt
\noindent
{\bf Exercise 20.}
Let $m$ be a positive integer, $K$ an algebraic number field
that contains a primitive $m$-th root of unity, and $J$ its
group of id\`eles. For a finite set $S$ of primes of $K$ we write
$$J_S=\prod_{\pp\in S}K_\pp^*\times\prod_{\pp\notin S}U_\pp,$$
$$K_S=K^*\cap J_S,$$
$$E_S=\prod_{\pp\in S}K_\pp^{*m}\times\prod_{\pp\notin S}U_\pp.$$
\par\noindent
(a)\enspace
Let $S$ be a finite set of primes of $K$ containing all infinite
primes and all primes dividing $m$, and suppose that $J=K^*J_S$.
Prove that the image of $K_S$ in $J_S/E_S$ is a group of
order $m^{\#S}$ isomorphic to $K_S/K_S^m$, and that the
quotient $J_S/(K_SE_S)$ is also of order $m^{\#S}$.
\par\noindent
(b)\enspace 
Let $J/J^m$ have the quotient topology from $J$.
Prove that the image of $K^*$ in $J/J^m$ is a {\sl discrete\/}
subgroup, isomorphic to $K^*/K^{*m}$. Prove that it can be identified
with the injective limit of the groups $K_S/K_S^m$, with $S$
ranging over all finite sets of primes of $K$.
\par\noindent
(c)\enspace 
Prove that the quotient group $J/(K^*J^m)$ is {\sl compact},
and that it can be identified with the projective limit of the groups
$J_S/(K_SE_S)$, with $S$ ranging over all finite sets of primes of $K$
for which $J=K^*J_S$.
\vskip 12pt
\noindent
{\bf Exercise 21.}
Let $K$ be an algebraic number field, and $\alpha$, $\beta$,
$\gamma\in K$.
\par\noindent
(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\noindent
(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
$\alpha x^2+\beta y^2=\gamma$.
\vskip 12pt
\noindent
{\bf Exercise 22.}
(a)\enspace
Prove that the field $\QQ(\sqrt5)$ has class number $1$,
and that the group of units of its ring of integers is
generated by $-1$ and $(1+\sqrt5)/2$.
\par\noindent
(b)\enspace 
Let $p$ be a prime number. Prove that there exists a
field $K$ satisfying
$$[K:\QQ]=4,\qquad\sqrt5\in K,\qquad|\Delta_{K/\QQ}|=25p$$
if and only if $p\not\equiv2$, $3\bmod5$.
Prove also that if such a field exists, it is uniquely
determined by $p$, up to isomorphism.
We denote this field by $K_{(p)}$.
\par\noindent
(c)\enspace
Prove that among all fields $K_{(p)}$, the only one that is Galois
over $\QQ$ is the field $K_{(5)}$. Can you embed
$K_{(5)}$ in a cyclotomic extension of $\QQ$?
\vskip 12pt
\noindent
{\bf Exercise 23.}
A number field is called {\sl totally real\/} if it has no
complex primes, {\sl totally complex\/} if it has no real
primes, and {\sl mixed\/} if it is neither totally real nor
totally complex.
The {\sl Fibonacci sequence\/} $(F_n)^\infty_{n=0}$ is inductively
defined by $F_0=0$, $F_1=1$, $F_{n+2}=F_{n+1}+F_n$.
\par\noindent
Let $p$ be a prime number with $p\equiv1$ or $4\bmod5$, and
let $K_{(p)}$ be as in Exercise 22.
\par\noindent
(a)\enspace
Prove that $K_{(p)}$ is mixed if and only if $p\equiv3\bmod4$.
\par\noindent
(b)\enspace
Suppose that $p\equiv1\bmod8$.
Prove that $K_{(p)}$ is totally real if $p$ divides $F_{(p-1)/4}$,
and totally complex otherwise.
\par\noindent
(c)\enspace
Suppose that $p\equiv5\bmod8$.
Prove that $K_{(p)}$ is totally complex if $p$ divides $F_{(p-1)/4}$,
and totally real otherwise.
\vskip 12pt
\noindent
{\bf Exercise 24.}
Let $p$ be a prime number with $p\equiv11$ or $19\bmod20$, and
let $K_{(p)}$ be as in Exercise 22.
Prove that the field $K_{(p)}$ has exactly one prime lying over $5$
if $p\equiv11\bmod20$, and exactly two primes lying over $5$ if
$p\equiv19\bmod20$.
\vskip 12pt
\noindent
Let $L$ be an algebraic number field of discriminant $\Delta$ over
$\QQ$.
{\sl Stickelberger's theorem\/} asserts that $\Delta\equiv0$ or $1\bmod4$
(Ludwig Stickelberger, Swiss mathematician, 1850-1936).
If $s$ denotes the number of complex primes of $L$, then the sign of
$\Delta$ is $(-1)^s$, so one can reformulate Stickelberger's theorem
by saying that $|\Delta|\equiv0$ or $(-1)^s\bmod4$.
In the Exercises 25--27 Stickelberger's theorem is
generalized to {\sl relative\/} discriminants.
\vskip 12pt
\noindent
{\bf Exercise 25.}
Let $E$ be a finite extension of $\QQ_2$, with ring of
integers $A$. Let $x\in E^*$. 
Prove that $E(\sqrt x)$ is unramified over $E$ if and
only if $x=(1+4y)z^2$ for some $y\in A$ and $z\in E^*$.
\vskip 12pt
\noindent
{\bf Exercise 26.}
Let $K\subset L$ be a quadratic extension of algebraic number fields,
and write $\Gal(L/K)=\{\id_L,\sigma\}$.
If $\aa$ is a non-zero ideal of the ring of integers $A_K$ of
$K$, we write $\NN\aa=\#(A_K/\aa)$.
\par\noindent
(a)\enspace
Let $\pp$ be a prime of $K$, and let $(-1)_\pp\in J_K$ be the
id\`ele that has coordinate $-1$ at $\pp$ and coordinate
$1$ at all other primes.
Prove that the image of $(-1)_\pp$ under the Artin map
$J_K\rightarrow\Gal(L/K)$ equals
\medskip
\halign{\hskip40pt#\hfil&\qquad#\hfil\cr
	$\id_L$&if $\pp$ is unramified in $L$,\cr
        $\sigma$&if $\pp$ is an infinite prime that ramifies in  $L$,\cr
        $\sigma^{(\NN\pp-1)/2}$&if $\pp$ is ramified in $L$ and
$|2|_\pp=1$.\cr}
\medskip\noindent
(b)\enspace 
Let $\dd_{L/K}$ denote the discriminant of $L$ over $K$, and
$s$ the number of infinite primes of $L$ that ramify over $K$.
Prove that $\NN\dd_{L/K}\equiv0$ or $(-1)^s\bmod 4$.
\vskip 12pt
\noindent
{\bf Exercise 27.}
Let $K\subset L$ be a finite extension of algebraic number fields,
and let $\dd_{L/K}$ denote the discriminant of $L$ over $K$.
Denote by $s$ the number of infinite primes of $L$ that
ramify over $K$.
Prove that $\NN\dd_{L/K}\equiv0$ or $(-1)^s\bmod 4$.
\vskip 12pt
\noindent
{\bf Exercise 28.}
Let $K$ be a number field, and $r$ the number of real primes
of $K$. Denote by $\Cl$ the class group of $K$, and by
$\Cl^*$ the {\sl strict\/} (or {\sl narrow\/}) class group of $K$;
i.e., the ray class group modulo the cycle
$\ff=\prod_{\pp\rm\;real}\pp$.
The $2$-rank $\rk_2A$ of a multiplicatively written abelian group
$A$ is defined to be the dimension of the $\Ftwo$-vector space $A/A^2$.
\par\noindent
(a)\enspace
Prove that $\rk_2\Cl\leq\rk_2\Cl^*\leq\rk_2\Cl+r-1$ if $r>0$.
\par\noindent
(b)\enspace 
Let $H=\{x\in K^*:K(\sqrt x)$ is unramified over $K$ at all
finite primes$\}$, and let $H^+$ be the set of elements
of $H$ that are positive at all real primes of $K$. Prove that
$\rk_2\Cl^*$ equals the dimension of the $\Ftwo$-vector space
$H/K^{*2}$,
and that $\rk_2\Cl$ equals the dimension of the $\Ftwo$-vector space
$H^+/K^{*2}$.
\vskip 12pt
\noindent
{\bf Exercise 29.}
Let the notation be as in the previous exercise.
\par\noindent
(a)\enspace
Let $a$, $b\in H$. Prove that
the norm residue symbol $(a,b)_{2,\qq}$ equals $1$
for every finite prime $\qq$ of $K$,
and that $\prod_{\pp\rm\;real}(a,b)_{2,\pp}=1$.
\par\noindent
(b)\enspace 
Prove that $\rk_2\Cl^*\leq\rk_2\Cl+[r/2]$.
\vskip 12pt
\noindent
{\bf Exercise 30.}
Let $p$ be a prime number, $E$ a finite extension of $\QQ_p$,
and $F$ a finite abelian tamely ramified extension of $E$.
Denote by $\pp_E$ and $\pp_F$ the maximal ideals of the rings of
integers of $E$ and $F$, and
by $k_E$, $k_F$ the residue class fields.
We let $I$ be the inertia group of $F$ over $E$, and $m=\#I$;
so $m$ is the ramification index of $F$ over $E$.
\par\noindent
(a)\enspace
Prove that the local reciprocity map $E^*\rightarrow\Gal(F/E)$
induces a surjective group homomorphism $k_E^*\rightarrow I$.
\par\noindent
(b)\enspace
Prove that $E$ contains a primitive $m$-th root of unity.
\par\noindent
(c)\enspace
{}From 254A we know that there is an injective
group homomorphism $I\rightarrow k_F^*$,
which is induced by the $k_F$-linear action of $I$ on
the one-dimensional $k_F$-vector space $\pp_F/\pp_F^2$;
so $\sigma\in I$ is sent to the image of $\sigma(\pi_F)/\pi_F$
in $k_F^*$, where $\pi_F$ is any prime element of $F$.

Prove that the composed map $k_E^*\rightarrow I\rightarrow k_F^*$
sends every $x\in k_E^*$ to $x^{-\#k_E^*/m}$.
\par\noindent
[{\sl Hint}. Prove first that the composed map sends a unit
$(u\bmod\pp_E)$ to the image of the norm residue symbol
$(\pi_E,u)_m$ in $k_F^*$, where $\pi_E$ is an arbitrary prime
element of $E$.]
\bye

