DETERMINATION OF THE HILBERT CLASS FIELD

FOR CERTAIN ALGEBRAIC NUMBER FIELDS ’

Thesis for the Degree Of Ph. D. f
MICHIGAN STATE UNIVERSITY
COLLEEN THEUSCH
1971

'.I"" t

This is to certify that the

thesis entitled

DETERMINATION OF THE HILBERT CLASS FIELD

FOR CERTAIN ALGEBRAIC NUMBER FIELDS
presented by

C . THEUSCH

has been accepted towards fulfillment
of the requirements for

PH . D . degree in MATHEMATICS

 

 

QEJ" ‘S\¥‘F\ \R'fl’“) Kr "‘x

Mdor professor

 

Date February L 1971

0-7639

 

ABSTRACT

DETERMINATION OF THE HILBERT CLASS FIELD
FOR CERTAIN ALGEBRAIC NUMBER FIELDS

BY

Colleen Theusch

To each algebraic number field K is
associated the Hilbert Class Field CF(K). This Field
CF(K) is characterized as the (unique) maximal
abelian unramified extension of K. CF(K)/K is of
degree h where h is the order of the ideal class
group of K, that is, h = h(K) is the class number
of the field K.

In general the determination of the Hilbert
Class Field of K is a very difficult problem. In
this thesis the properties of the number discriminant
of an element are employed to explicitly determine
the Hilbert Class Field of some quadratic number
fields. Then localization techniques are used to
determine the Hilbert Class Field of certain BEE:
normal algebraic number fields. A characterization
of the pure fields .(fields of the form Q(n/a) )
for which these techniques are valid is given through

the theorems and corollaries.

DETERMINATION OF THE HILBERT CLASS FIELD

FOR CERTAIN ALGEBRAIC NUMBER FIELDS

BY

”I, .j) '1 "5

Colleen’Theusch

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1971

ACKNOWLEDGMENTS

The author thanks her adviser, Professor
Charles R. MacCluer, for his suggestions in the
preparation of this thesis. She acknowledges useful
discussions with various members of the department
of mathematics, particularly with Professor Robert
Spira. She thanks Edith Stern for her aid in
proofreading. Finally, she wishes to express her
gratitude to Philip Pfaff for his encouragement,

understanding, typewriter and other arrangements.

TABLE OF CONTENTS

PAGE
ACKNOWLEDGMENTS O O O O O O O O O O O O O O O O i i
INTRODUCTION . . . . . . . . . . . . . . . . . 1

CHAPTER
I. BACKGROUND MATERIAL . . . . . . . . . . 3
II. DISCRIMINANTS DETERMINE CLASS FIELDS. . 16

III. HILBERT CLASS FIELDS THROUGH
LOCALIZATION TECHNIQUES . . . . . . . . 23

BIBLIOGRAPHY O O O O O O O O O O O O O 0 O O O 37

iii

INTRODUCTION

To each algebraic number field K is
associated the Hilbert Class Field CF(K). This field
CF(K) is characterized as the (unique) maximal
abelian unramified extension of K. CF(K)/K is of
degree h where h is the order of the ideal class
group of K, that is, h = h(K) is the class number
of the field K.

In general the determination of the class
field of K is a very difficult problem. The theory
of complex multiplication presents an adequate, though
rather involved, analytic method for determining the
class fields of the imaginary quadratic fields. When
K/Q is cyclic of prime degree n, CF(K) may be
abelian and hence identical with the genus field of
K, the maximal abelian subfield of CF(K)/Q. Thus
the method used to determine the genus field as found
in [2] applies. However, even in these cases the
explicit determination of the Hilbert Class Field is
difficult since the necessary calculation of the
class invariants and automorphisms is frequently

extremely tedious.

In the second chapter of this thesis I
will employ the properties of the number discriminant
of an element to explicitly determine the Hilbert
Class Field of some quadratic number fields. Local-
ization techniques which can be used to explicitly

determine the Hilbert Class Field of certain non-normal

 

number fields are presented in the third chapter. A
characterization of the pure fields (fields of the
form Q(n/a) ) for which these techniques are valid

is given through the theorems and corollaries.

CHAPTER I

BACKGROUND MATERIAL

In this chapter we will consider certain
concepts which are basic to the understanding of the E
material in Chapters II and III. I
We will be concerned here with algebraic
number fields, that is, finite algebraic extensions
of Q, the field of rational numbers. Any such
extension K of Q can be obtained by adjoining to
Q the root a of an irreducible monic polynomial
f(X) in Q[X]. The algebraic number a is referred

to as a primitive element for the extension while

 

f(X) = O is called the defining equation of a

 

which is denoted by f(X) = Irr.(a,Q). These two
concepts are defined similarly for an arbitrary
algebraic number field K = Q(a) so that we have an
extension L = K(B) with f(X) = Irr.(8,K) in K[X].
To retain the notion of unique factorization
in the algebraic number field K one considers the
ideals of K rather than its elements. Following
common practice we shall refer to a prime ideal of K

simply as a finite prime.

Ramification of the primes of K in
extensions of K plays a crucial role in class

field theory.

Definition l. A finite prime P of a

 

field K ramifies in an extension L of K if it
has a repeated factor in L, that is, if P extends

er

e . . .
to Q1 -°°Qn n Wlth each Qi prime in L and for

which at least one e- is greater than 1.

1

We state Kummer's Theorem here since it
supplies one method for determining the factorization

of all finite primes P of K in L = K(B)-

Kummer's Theorem. Let R be a Dedekind

 

Domain with quotient field k, K a finite separable
extension of k of degree n, and let S = intRK

be the integers of K. Suppose K has an integral
basis l,6,...,6n- over R. Let f(X) = Irr.(6,k)
and for a prime ideal P of R suppose

f(X) 2 flel(X)’°-fgeg(X) (mod P) where the fi(X)
are all distinct monic R-polynomials which are
irreducible modulo P. Then in S, P has the prime

. . . e1 99
ideal factorization SP = Q1 ---Q9 where

Q.

l = (SP: fi(e))o

Consideration of some valuation theory will
enable us to understand the ramification of the finite

and infinite primes of K.

Definition 2, A valuation of the field K

 

 

is a function ¢ from K into the non-negative reals
such that
(i) ¢(a) = 0 if and only if a = 0
(ii) ¢>(ab) = ¢(a)¢(b)
(iii) There exists a real constant C such

that ¢(a) :_1 implies ¢(l+a) i C.

An equivalent condition to (iii) is

(iiia) ¢(a+b) :_¢(a) + ¢(b). [6].

Definition ;. II¢H = inf. C where C runs

 

over all constants of Definition 2, (iii) above is

called the norm of o.

Each valuation on a field determines a
Hausdorff topology on that field. Those valuations
which yield the same topology are considered
equivalent and thus all valuations can be divided into
equivalence classes. For convenience we shall refer
to a complete equivalence class of valuations as a

valuation. The equivalence classes are sometimes

also called prime divisors of the field K and thus
each valuation ¢:K + R is associated with some
prime divisor or prime P of K. If “¢" = l we

term o a nonarchimedian valuation and the associated

 

 

P is then a nonarchimedian or finite prime. On the

-

other hand when H¢H > 1, ¢ and p are called

 

 

archimedian or infinite. We are primarily concerned

 

with the latter here since we already have a method
for determining the ramification of the finite primes
in a given field.

Archimedian valuations are in a sense an
extension of the concept of absolute value in a
complex field. For consider any isomorphism 0(i) of
K into the complex number field C. Galois theory
assures us that there are exactly n such isomorphisms
where n is the degree of K over Q. The image
field of say r of these isomorphisms will be
contained in the field of real numbers. The remaining
n-r occur in s conjugate pairs and have complex
image fields. Clearly n = r+25. We set

¢(l)(x) = [0(1)(x)l and note that this yields r+s

(i)

distinct valuations. As one would expect, the ¢

which correspond to those 0(1) which have real

images are termed real valuations, the others being
(i)

(I)

 

called complex valuations. The infinite primes P

 

are termed real or complex in accord with their

(1)
associated real or complex valuation ¢ .
1
Now if ¢( ) is a real valuation of K
which when extended to L becomes complex, we say

i
that its associated prime Poo has ramification

 

index ei = 2. In all other cases, that is, when
¢(i) remains real when extended or when ¢(i)
complex in K, the ramification index of the
associated Pm(i) is 1- A fact that will be used
repeatedly in what follows is that when K/k is

unramified, so is KL/kL.

While discussing ramification we referred
to class field theory. In the following pages we
will be particularly interested in the Absolute or

Hilbert Class Field of an algebraic number field

 

K = Q(a).

'Definition 3. The Hilbert Class Field of K

 

 

is the (unique) maximal abelian unramified extension
of K and is denoted by the symbol CF(K), where
unramified means unramified with respect to both
finite and infinite primes. Such an extension CF(K)

exists as demonstrated by Furtwangler in 1903. [4].

Let I be the group of all fractional ideals

and H the subgroup of principal ideals in K. The

order of I/H, the ideal class group of K, is

 

called the class number of K denoted h(K). One of

 

the standard facts of class field theory is that the
degree of the extension CF(K) over K is equal to
the class number of K.

The class number is an indication of how
far the ring int K (the integers of K) is removed
from being a principal ideal domain. In fact, h(K) is
1 if and only if int K is a principal ideal domain,
in which case K is its own Hilbert Class Field. It
is well known that while every ideal J of K when
lifted to CF(K) becomes principal, CF(K) itself
frequently contains non-principal ideals. This
gives rise to the classical problem of class field
towers. That is to ask, is the chain
K CKICK

o
where each Kn is the Hilbert Class Field of K

Coooc C000
2 Kn '

n-l'

necessarily finite? Clearly an affirmative answer to
this question would imply that each such tower would
have a principal ideal domain as one of its terms.
However, this question was answered negatively by
Golod and Safarevic in 1964. [5].
Generally it is extremely difficult to

calculate the Hilbert Class Field of an algebraic
number field. One criterion that may be used is that

the Galois group G(CF(K)/K) is isomorphic to the

ideal class group I/H. Thus in those cases where
this group has already been determined, considerations
of extensions of K having this as Galois group may
aid in the determination of the Hilbert Class Field

of K, since if such an extension is found to be
unramified, the uniqueness of CF(K) insures the
desired result.

Nevertheless, several inherent difficulties
remain. Even when the composition of the ideal class
group and consequently the class number is known, the
actual production of the abelian unramified extension
is problematic. Moreover for a wide range of fields
even the class number itself has not yet been computed.
Several tables of class numbers for quadratic, cubic,
and cyclotomic fields are exhibited by Borevich and
Shafarevich in [3]. All class numbers quoted in
this thesis are taken from that reference.

As has already been indicated, the problem
of calculating class fields involves proving that no
prime of K can ramify in certain extensions under
consideration. One method of achieving this is
through the use of discriminants.

Let L = K(e) be a finite separable
extension of K and let 01'02"°"On denote the n
distinct K-isomorphisms of L into the algebraic

closure of K. Let ej = o,(e), and
J

n-l 2 n-l 2 2
dL/K(l,6,...,e ) — l e e ... e — iI<Ij(ei-ej)

 

 

n-l)

Definition g. The element d (l,6,...,6

L/K

will be called here the number discriminant of 8.

 

 

Discriminants are of importance in
determining class fields since, according to Dedekind,
any finite prime P of K that ramifies in L must

n-l)

divide every number discriminant dL/K(l,6,...,6
Thus in searching for unramified extensions L of K
it is sufficient to show that there exist relatively
prime number discriminants for two elements of L or
to prove that those primes of K which divide any
given number discriminant remain unramified in L.
Both these methods will be employed in the following
chapters.

We must consider cyclotomic fields since
a cyclotomic extension of some algebraic number fields

will yield the Hilbert Class Field CF(K).

Definition E. An extension of the rational

 

field obtained by adjoining a primitive n-th root

of unity is termed a cyclotomic number field.

 

11

Definition 1. K(a) is a cyclotomic

 

 

extension of K if a is a primitive element for a

 

subfield of a cyclotomic number field.

Let be a primitive n—th root of unity.

Cn
If K is an algebraic number field that meets QICn)

at Q, then K(cn) is an extension of degree ¢(n)
over K. In particular, for n = p (a prime) and any
K which does not contain any subfield of Q(cp) other
than Q, (K(Cp):K) = p-l. While every cyclotomic

number field is abelian, only those for which n = 2, 4,

e

p , and 2pe

with p an odd prime are cyclic.
The defining equation for K(Cp) is
Xp"l + XE)"2 + ... + X + l = 0. If k is a subfield

of Q(Cp) with index m, then k = Q(6p) where 6p
a a2 0 O 0 am-1
p + 2;p + C9 + + C9 '

2 i a < p, and conversely. For instance Q(Cp+Cp'l)

is of the form C

has index 2 in Q(Cp). Since Eb = Cp'l, §p+;p'l

and all its conjugates are real. Thus Q(cp+§p'1)

is real as well as all its subfields. Since Q(Cp)

is cyclic, Q(Cp+cp‘l) is its only subfield of

index 2. Thus all the subfields of Q(§p) of odd
degree are contained in Q(;p+§p‘1) and hence are

real. When p E 1 (mod 4), Q(/p) is contained in

Q(Cp) and /p has defining equation X2 + X - (p-l)/4=0,

while for p E 3 (mod 4), Q(/-p) is the quadratic

12

subfield of Q(Cp) with defining equation

x2 + X + (p+1)/4 = 0.

Definition g. Let f(X) = 0 be the

 

defining equation of the integer B of L = K(B)
over K. The integer f'(8) of L is called the

number different of 8 over K.

 

The number discriminant of an integer is
(within a sign) the norm of the number different.
Further any prime P of L which is ramified over
K divides every number different. Other useful
facts are that

(9-1)

2I = (-1) 2 pP‘2

d(1,cp,...,cpp' ,
and that f'(Cp) divides p.

Thus the only primes P of L = K(;p)
which can ramify over K must divide p. Therefore
when H = K(6p) is a subfield of L, the only primes
of H which can ramify over K are those primes
which are divisors of p, since if a prime ramifies
in a subfield of L it must also ramify in L.

Since localization methods will be used to
exhibit the Hilbert Class Field for some algebraic
number fields, we consider some local field theory ---

primarily that related to Qp, the completion of the

field Q of rational numbers with respect to a

l3

p-adic valuation.

Definition 2. Let x be any rational

 

number other than zero. Then for any prime p,
x = pa(a/b) where p divides neither of the

integers a or b. Let

IXIp = 0 if x = 0
p‘a if x # 0.
The non—negative real valued function |x| + |x| is

P
called the p-adic valuation on Q.

 

 

The function thus defined is clearly a
valuation. Moreover for a p-adic valuation
condition (iiia) may be replaced by the stronger
(iiib) |a+b|p :_max (Ialp,|b|p).
A p—adic valuation is non—archimedian as can be seen
from (iiib) which yields Ia+l|p :_max(|a|p,|l|p) = 1
when la] 1 l. The ring of integers of Qp is

P
denoted by Op = {a in Qp:lalp : 1} while
P = {a in Qp:|a|p < l} is the unique prime ideal of
Qp. Since Qp has only one prime ideal, consideration

of the ramification which may occur in Qp(a) is much

simpler than that in an algebraic number field.

Definition lg. Qp(a)/Qp is totally ramified

 

 

if P, the unique prime ideal of Qp, has ramification

14
index in Qp(a) equal to the degree of the extension.
From (x-1)¢pIX) a (x-i)P (mod p) it
follows that ¢p(x) E (X-l)p'1 (mod p) where ¢p(X)

is the p-th cyclotomic polynomial and hence Qp(cp)

is totally ramified by Kummer's Theorem.

Definition ll. Qp(a)/Qp is tamely ramified

 

if p does not divide the degree of the extension.

Since the degree of Qp(cp) over Qp is

p-l, Qp(z;p)/Qp is tamely ramified.

We state one form of Hensel's Lemma.

 

Let R be a Dedekind Domain with quotient
field k and let P be an integral prime ideal of
R. Suppose that f(X) is a polynomial of R[X]
such that for some a in R, ,f(a) E 0 (mod P) and
f'(a) ? 0 (mod P). Then f(X) = 0 has at least one

(non—repeated) root in the local field kP.

Hensel's Lemma implies that Qp contains
the (p-l)-th roots of 1 since
xP‘l-i 2 (X-1)(X—2)...(X-(p-l)) (mod p)
so that ¢p_l(X) must split completely in Qp.

For our purpose one of the most crucial

15

aspects of local field theory is that the ramification
index of a rational prime p of Q in an extension
0(a) is the same as the ramification index of the
unique prime ideal P of Qp(a) as has been shown

by Artin.

CHAPTER II

DISCRIMINANTS DETERMINE CLASS FIELDS

In this section we shall see that the class
field of Q(/m), for certain composite m, can be
determined solely by consideration of relative
number discriminants. We will list those fields with

|m| < 500 to which the particular propositions apply.

Theorem I, Let m = pq > 0 where p and
q are distinct positive prime numbers with

1 (mod 4), and suppose Q(/m) has class number

"O
n

h = 2. Then

CF(Q(/m)) = QI/p,/q).

Proof. We have already seen that a

necessary condition that a finite prime of K = Q(/m)

 

ramify in an extension of K is that it divide
every relative number discriminant. Clearly

L = Q(/p,/q) is an extension of K of degree 2.
Further (l+/p)/2 and either (l+/q)/2 or /q

(depending on whether or not q E 1 (mod 4)) are

16

17

integral. Now we have the relative number discriminants

for L/K:
d(l, (l+/p)/2) = p
d(l, /q) = 4q
d(1, (1+/q)/2) = q-

Since p and 4q are relatively prime, it is
impossible for any finite prime of K to ramify in the
extension L. Moreover since L is real, the
archimedian primes of K also are unramified. Hence
Q(/p,/q) is unramified of degree h = 2 over Q(/m)
and thus must be the Hilbert Class Field of

K = Q(/m).

This theorem can be applied to a rather
long list of real quadratic number fields. When
Hp E 1 (mod 4) with m = pq, Q(/m) has class number
2 for the following m and hence for these m we

have CF(Q(/m)) = Q(/p,/q).

10
15
26
34
35
39
51
55
58
65
74
85
87
91
95
106
111
115
119
122
123
143
146
155
159
178
183
185
187
194
202
203
205
215
218
221
247

13
17

13
17

29
37

29
13

53
37

17
61
41
13
73

53
89
61

17
97

..-
O
...:

29

109
13
13

WP

I.» \D \l

H

\JNNI—‘WWNWWNHUJNQNUJNHQWHNHNHWWQNNNN s-Q
I—N l-'

bk
(AH

I—II-Ito
\DQ

18

259
265
267
287
295
298
299
303
305
314
319
327
335
339
355
362
365
371
377
386
391
394
395
403
407
411
415
447
451
458
466
471
481
482
485
493

Pg

37

5
89
41

149
13
101

157
29
109

113
181

53
13
193
17
197

13
37
137

149

41
229
233
157

13
241

19

7
53
59
23
61
11
67
71
73
29
23
79
31
11
83

ll

37

97
29

19

Modification of the conditions placed on m
yield several corollaries to the above theorem. The

first of these concerns imaginary quadratic fields.

Corollary 1} Let m = pq > 0 where p and

 

q are distinct positive prime numbers with p # 2 and

such that Q(/—m) has class number 2. Then

CF(Q(/—m))

Q(/p./-q) if p 1 (mod 4)

CF(Q(/-m)) Q(/-p./q) if p

3 (mod 4).

Proof. The finite primes of Q(/-m) other

than p cannot ramify in the extension since we have

d(1, (i+/p)/2) p if p 1 (mod 4)

d(l, (i+/-p)/2 = -p if p 3 (mod 4).

But neither can the (repeated) prime factor of p in
Q(/—m) ramify in the extension since p # 2 and
d(1, /q) = 4g. The archimedian primes of Q(/-m)

are already complex and hence are no cause for

concern. The result follows as in the theorem.

As examples of this we have
CF(Q(/-m)) = QI/p./-q) or QI/-p,/q) for the

following m.

20

m p q m p q
6 3 2 91 7 13
10 5 2 115 5 23
15 3 5 123 3 41
22 11 2 187 11 17
35 5 7 235 5 47
51 3 17 267 3 89
58 29 2 403 31 13

427 7 61

Returning again to the case of the real

quadratic fields we have

Corollary 2. Let m = pqr > 0 where p,

 

q, and r are distinct positive prime numbers with
p E 1 (mod 4) and neither q nor r E! 1 (mod 4).

Suppose that the class number of Q(/m) is 2. Then

CF(Q(/m)) = Q(/p./qr).

Proof. For the integers (l+/p)/2, /qr,
and (l+/qr)/2 of Q(/p,/qr) we have the relative
number discriminants over Q(/m) equal to p, 4qr,
and qr respectively. Since p and 4qr are
relatively prime and since Q(/p,/qr) contains only
real infinite primes, Q(/p,/qr) is an unramified
extension of degree 2 over Q(/m) and hence its

Hilbert Class Field.

As a result of this corollary we have

21

CF(Q(/m)) = Q(/p,/qr) for the following m.

m p q r m p q r

30 5 2 3 285 5 3 19
70 5 2 7 286 13 2 11
78 13 2 3 310 5 2 31
102 17 2 3 318 53 2 3
105 5 3 7 345 5 3 23
110 5 2 11 357 17 3 7
165 5 3 11 366 61 2 3
174 29 2 3 374 17 2 11
182 13 2 7 385 5 7 11
190 5 2 19 406 29 2 7
222 37 2 3 429 13 3 11
230 5 2 23 430 5 2 43
238 17 2 7 465 5 3 31
246 41 2 3 470 5 2 47
273 13 3 7 494 13 2 19

Further changes in the hypotheses of the

theorem give the final result of this section as

Corollary 3. Let m = 2pq > 0 where p

 

and q are distinct odd positive primes with
p,q E 3 (mod 4). Moreover let the class number of

Q(%m) be 2. Then

CFIQI/mI) = QI/2./pq).

Proof. Since p,q E 3 (mod 4) we have
pg 5 1 (mod 4). Thus (l+/pq)/2 and /2 are
integers of Q(/2,/pq) with d(1, (l+/pq)/2) = pq

while d(1, /2) = 4. Moreover all archimedian

22

primes remain real. Hence the conclusion follows

immediately.

Since the class number of Q(/m) is 2
for the following m, we have CF(Q(/m)) = Q(/2,/pq)

for these m.

m 2 p q m 2 p q
42 2 3 7 282 2 3 47
66 2 3 11 354 2 3 59

114 2 3 19 402 2 3 67
138 2 3 23 418 2 11 19
154 2 7 11 426 2 3 71
186 2 3 31 474 2 3 79
258 2 3 43 498 2 3 83
266 2 7 19

ll AlIilIlI. I. I ‘
Ill I‘ll-INN

CHAPTER III
HILBERT CLASS FIELDS THROUGH

LOCALIZATION TECHNIQUES

We now turn our attention to a more
interesting class of fields --- the pure fields,
that is, those of the form Q(n/a). Since the
rationals do not contain the n-th roots of unity
for n # 2 the pure fields are never normal over
the rationals when n f 2. In order to determine
the class fields of certain of these we will employ
localization techniques. The lemmas that follow are

the foundation of the technique.

Lemma A. Let p be a positive rational

prime number and C a primitive p-th root of

P
= P‘1 _
unity. Then Qp(cp) Qp( / p) where Qp denotes
the field of p-adic rational numbers.
Proof. Since Qp(2;p)/Qp is tamely and
totally ramified we have that Qp(cp) = Qp(p"1/6p)

for some unit 8 of Qp. But in fact, -p = —u(1_;)P'1

where -u E -(p-l)! E l mod(l-C). But then by

23

24

Hensel's Lemma there exists a unit a in Qp such

-1
that 8p = -u.

 

-1
Corollary. Qp(p /-p)/Qp is cyclic, and

tamely and totally ramified.
Proof. Q contains the (p-l)-th roots

P
of unity.

EEEEE.§° If h is an odd divisor of p-l,
then Qp(h/p) = Qp(6) where 6 is a primitive
element for the h-th degree subfield of Q(;p).

Proof. Clearly Qp(h/p) = Qp(h/-p) since
h is odd. Qp(cp) is a cyclic extension of Qp
and thus contains only one subfield of degree h.

This, together with Lemma A, yields the conclusion.

We are now in a position to prove our first
theorem concerning some fields which are ngt_normal
over the rationals. Direct application of this
theorem will enable us to actually display the Hilbert
Class Field of certain fields of this type for which

the class number is known.

Theorem 2. Let the class number h of

Q(e/p) be odd and let h be a divisor of e. Then

if p E 1 (mod e) we have

25

CF(QIe/pI) = 0(e/p,6)

where 0 is a primitive element for the unique
subfield of degree h of the cyclotomic field Q(cp).
Proof. Since the discriminant of the

number over Q is a power of p, it is clear

Cp
that the only finite prime of Q(e/p) that can
ramify in the extension Q(e/p,§p) and hence in
Q(e/p,8) is e/p. But not even e/p can ramify.
For by Hensel's Lemma, Qp contains the (p-l)-th
roots of unity and hence also the e—th roots of
unity. Thus, locally, Qp(e/p)/Qp is normal, in
fact, cyclic. Since h is odd and divides e,
Qp(e/p) contains Qp(h/p). On the other hand,
Qp(h/p) = Qp(0) by Lemma B. The fact that global
ramification can be determined by local ramification
indicates that the ramification index of p in
Q(e/p,9) as well as in Q(e/p) is e. Hence e/p

does not ramify in the given extension.

Consideration of the following diagram

should help clarify the preceding statements.

26

QPIP‘l/p)

 

h h
Qp( /p) = QPI /-p)

 

 

 

p-l_ _
Qp( / p) - Qp(cp)

 

e—
Qp( / p)

= Qp(9)

27

Now that we have disposed of the finite

. . . . . . . i
primes we w111 conSider the infinite primes Pm( )

in Q(e/p). The only Pm(i) which could ramify in
the extension are those associated with the real
archimedian valuations ¢(i) of Q(e/p). Now since
Q(6)/Q is galois and of odd degree h, 0(0) is a
real field. Hence 6 and all its conjugates are also
(i)

real. Thus the extensions of the real 0 are real

in Q(e/p,0), and hence none of the Pm(1) ramify.
Thus Q(e/p,0) is an abelian unramified
extension of Q(e/p) of degree h and is therefore

the Hilbert Class Field.

We have now arrived at a position from which
we can specify the Hilbert Class Field for specific
pure fields. Since k = Q(3/p) has class number 3
for p = 7, 13, 19, 31, and 37 [3] the Hilbert Class
Field of k is k(0) where 0 is a primitive
element for the cubic subfield of Q(Cp). In

particular we have

CF(Q(3/7))=Q(3/7.C7+C76)
CF(Q(3/13))=Q(3/13,§ +2 5+; °+g 12)

13 l3 13 13
CF(Q(3/19))=Q(3/19.c19+47+c°+c‘1+412+c‘°)
CF(Q(3/31))=Q(3/31,C31+CZ+C“+C°+C15+C16+C23+C27+C29+c3°)
CF(Q(3/37))=Q(3/37,§ 7+cs+§°+§1°+§11+§1“+§23+;25+§27+

3

C29+C31+C36)O

28

We note the corresponding defining equations

which yield these extensions.

Irr(8,Q(3/7 )) = x3 + x2 - 2x - 1
Irr(8,Q(3/13)) = x3 + x2 - 4x + 1
Irr(8,Q(3/19)) = x3 + x2 — 6X - 7
Irr(0,Q(3/3l)) = x3 + x2 - 10x - 8

Irr(e,Q(3/37)) = X3 + x2 — 12x + 11
We continue with

Lgmm3_g. If e is even and p E 1 (mod 2e),
then Qp(e/p) = Qp(e/-p).

Proof. We have already seen that Qp
contains the (p-l)—th roots of unity. Thus under
the hypotheses, Qp contains the 2e-th roots of
unity and hence also the e-th roots of -l. The
conclusion is immediate.

Lemma 2. Qp(C ) contains Qp(e/p) when

P
both e is even and p E 1 (mod 2e).
Proof. We first note that since
p E 1 (mod 2e), Qp(p-l/-p) contains Qp(e/—p). Then
Lemma C followed by Lemma A yields
9/ = e/— = e c
QPI p) Qp( p) Qp( )<: Qp( p)

where 0 is a primitive element for the e-th degree

29

subfield of Q(cp).
The preceding lemmas enable us to prove
Theorem 3. Let p E 1 (mod 2e) and suppose

Q(e/p) has class number h with e z 0 (mod 2h).

Then
CF(Q(e/p)) = QIe/p,e)
where 0 is a primitive element for the h-th degree

extension of Q(/p) in Q(Cp).

Proof. The stated hypotheses imply that

p 1 (mod 4). Since Q(Cp) contains Q(/p) when

1 (mod 4), Q(e/p,6) is an extension of Q(e/p)

P

of degree h.

QIe/pIGI
/ e/2
Q(e/p) Q(/p,8)
e/2 h
Q(/p)
2

30

Clearly e/p is the only finite prime of Q(e/p)
that can possibly ramify in Q(e/p,6). To see that
this in fact cannot occur, we again localize at p.
From Lemma D we have that Qp(cp) contains Qp(e/p)
with the unique subfield Qp(/p,6) = Qp(2h/p) of
degree 2h. Since e E 0 (mod 2h), Qp(e/p) contains

Qp(/p,0).

The following diagram illustrates the above

containments.

= p-1 -
Qp(cp) Qp( / p)

(p-l)/2 = 2m

 

Qp(e¢p) = Qp(e/-p)'
l em
Qp(2h/p) = Qp(/p,6)
h
Qp(/p)
2

 

31

The local ramification index e of p in
Qp(e/p) indicates that the global ramification index
of p in Q(e/p,6) must also be e. Hence e/p
does not ramify in the extension.

Since the degree of Q(§p)/Q(8) is even,
0(6) is real. Thus all the conjugates of 8 are
real, and hence the infinite primes of Q(e/p) have
ramification index 1 in Q(e/p,0). Thus the Hilbert

Class Field of Q(e/p) is indeed Q(e/p,0).

Clearly Theorems 2 and 3 are also valid for
Q(e/pa) where a and e are relatively prime since
in that case Q(e/p) = Q(e/pa).

When p E 1 (mod 4), Q(§p) does ngt_
contain Q(/-p). Thus the hypothesis e E 0 (mod 2h)

of Theorem 3 can be omitted to obtain

Corollary 1, Let p E 1 (mod 2e) with

 

e E 0 (mod 2), and suppose that Q(e/—p) has class

number h where h divides e. Then
CF(Q(e/-p)) = QIe/—p,e)
where 0 is a primitive element for the h-th degree

subfield of Q(Cp).

Proof. Since e is even the archimedian

32

primes in Q(e/-p) are already complex and therefore
cannot further ramify. Moreover the hypotheses

imply that p E 1 (mod 4). Since p E 1 (mod 2e),
Qp(e/-p) is cyclic over Qp by Lemma C. Application
of Lemma A yields Qp(6) = Qp(h/—p) as the unique

subfield of Qp(e/-p) of degree h.

As a special application of the above we

have

Corollaryg. Let p E 1 (mod 4) be such

 

that Q(/~p) has class number 2. Then

CF(Q(/—p)) = QI/-p./p) = QIi./p).

Proof. Q(/p) is the quadratic subfield

of Q(§p) when p E 1 (mod 4).

For p = 5, 13, and 37, Q(/-p) has class

number 2. Hence

CF(Q( /—5)) = QIi. /5)
CF(Q(/-13)) = Q(i,/l3)
CF(Q(/-37)) = QIi,/37).

The radicands of the pure fields we have

considered up to this point have all been primes.

33

The theory can be extended to include certain
composite radicands. To begin this we shall deal

with cubic extensions of the rationals.

Lemma E, Let a,b be in Qp* (the
multiplicative group of the non-zero elements of
Qp) but not in Qp*3. Then when a/b is an element
*3 3 = 3
of Qp . Qp( /a) Qp( /b).

Proof. Clear.

Lgmma_§. Let the rational integer r be a
cubic residue modulo p with p a positive rational
prime distinct from 3. Then Qp(3/p) = Qp(3/rp).

Proof. By Hensel's Lemma, r is in Qp*3

and rp/p = r.

1 (mod 3) and let r

Ill

Theorem 4. Let p
be a cubic residue modulo p. Suppose h(Q(3/rp)) =3.

Then

CF(Q(3/rp)) = Q(3/rp.6)

where 6 is a primitive element for the cubic
subfield of Q(Cp).
Proof. As we have observed before, 3/p is

the only finite prime that could ramify in the

34

extension Q(3/rp,0) of Q(3/rp). Localizing at p
we see that since p E 1 (mod 3), Qp contains the
cube roots of unity. Hence Qp(3/P)/Qp is cyclic
and from Lemma F it follows that Qp(3/rp)/Qp is also
cyclic since r is a cubic residue modulo p. Moreover,
application of Lemmas F and B yield. Qp(3/rp) = Qp(6).
Thus globally p has ramification index 3 in
Q(3/rp,9) which is also its index in Q(3/rp).

Since Q(0)/Q is galois and of odd degree,
Q(6) is a real field. Therefore the real archimedian
primes of Q(3/rp) extend to real primes in
Q(3/rp,9). Hence Q(3/rp,9) is an abelian unramified
extension of Q(3/rp) of degree 3 and is therefore

the Hilbert Class Field.

For instance note that Q(3/42) has class
number 3. Thus CF(Q(3/42)) = Q(3/42,6) where 6
is a primitive element for the cubic subfield of
Q(Cp). That is, CF(Q(3/42)) = 0(3/42,c7+c7‘), with

defining equation Irr(6, Q(3/42)) = X3 + X2 - 2X - 1.
It can readily be seen that in Lemmas E and
F and in Theorem 4, the prime 3 can be replaced by

any odd q # p so that we have

Theorem 43. Let p E 1 (mod q) and

35

q E 1 (mod 2). Further let r be a q—th power
residue modulo p, and suppose that h(Q(q/rp)) = q.

Then
q q
CF(Q( /rp)) = Q( /rp,e)

where 0 is a primitive element for the q-th degree

subfield of Q(Cp).

A very slight modification in the hypotheses

yields

Theorem 42, Let p E 1 (mod 2g) with
q E 0 (mod 2) and let the positive rational integer
r be a q-th power residue modulo p. Suppose that

h(Q(q/-rp)) = q. Then
CF(Q(q/-rp)) = QIq/—rp.e)

where 0 is a primitive element for the q-th degree
subfield of Q(Cp).

Proof. Since r is a q—th power residue
we have Qp(q/-rp) = Qp(q/—p), while the first
corollary to Theorem 3 assures us that
Qp(q/-p) = Qp(q/-p,0). Hence the finite primes of

Q(q/-rp) do not ramify in Q(q/-rp,0). Since the

36

archimedian primes of Q(q/-rp) are complex, no

ramification can occur.
Similarly we have

Theorem 43, Let p E 1 (mod 2q) with
q E 0 (mod 2) and let r > 0 be a q—th power

residue modulo p. Suppose that h(Q(q/rp)) = q. Then
CF(Q(q/rp)) = QIq/rp,e)

0 as before.

Proof. We need only note that since
p E 1 (mod 2q) Qp contains the q-th roots of —1.
Thus Qp(q/rp) = Qp(q/-rp) = Qp(q/—p) = Qp(q/—p,0).
The last equality is a result of Lemma A. Regarding
the infinite primes, we need only observe that since

(p-l)/q is even, Q(6) must be real.

BIBLIOGRAPHY

BIBLIOGRAPHY

Bachman, George, Introduction to p-adic
Numbers and ValuatIOn Theory, Academic Press,
New York, 1964.

 

 

Borel,A., Chowla,S., Herz,C.S., Iwasawa,K., and
Serre,J-P., Seminar gn_Complex Multiplication,
Springer-Verlag, Berlin, 1966.

Borevich,Z.I., and Shafarevich,I.R., Number
Theory, Academic Press, New York, 1966.

Furtwangler, Philip, "Allgemeiner Existenzbeweis
fur den Klassenkorper eines beliebigen
algebraischen Zahlkorpers", Mathematische
Annalen, 63, 1909, pp.l-37.

 

Golod,E.S., and Safarevic,I.R., "On Class Field
Towers" (Russian), Izv. Akad. Nauk. SSSR 28,
1964, pp.26l-272, English translation in Am,
Math. Soc. Transl. (2) 48, pp.91-102.

 

 

 

 

Weiss, Edwin, Algebraic Number Theory,
McGraw-Hill Book Company, New York, 1963.

 

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