Fermat's last theorem: a genetic introduction to algebraic number theory
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
2000
|
| Ausgabe: | 1. softcover printing |
| Schriftenreihe: | Graduate texts in mathematics
50 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 403 - 407 |
| Beschreibung: | XV, 410 S. 25 cm |
| ISBN: | 0387950028 |
Internformat
MARC
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| 084 | |a MAT 120f |2 stub | ||
| 100 | 1 | |a Edwards, Harold M. |d 1936-2020 |e Verfasser |0 (DE-588)117709700 |4 aut | |
| 245 | 1 | 0 | |a Fermat's last theorem |b a genetic introduction to algebraic number theory |c Harold M. Edwards |
| 250 | |a 1. softcover printing | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 2000 | |
| 300 | |a XV, 410 S. |b 25 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate texts in mathematics |v 50 | |
| 500 | |a Literaturverz. S. 403 - 407 | ||
| 650 | 4 | |a Algebraic number theory | |
| 650 | 4 | |a Fermat's last theorem | |
| 650 | 0 | 7 | |a Algebraische Zahlentheorie |0 (DE-588)4001170-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Fermatsche Vermutung |0 (DE-588)4154012-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Zahlentheorie |0 (DE-588)4067277-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Fermatscher Satz |0 (DE-588)4325816-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Fermatsche Vermutung |0 (DE-588)4154012-8 |D s |
| 689 | 0 | 1 | |a Algebraische Zahlentheorie |0 (DE-588)4001170-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Algebraische Zahlentheorie |0 (DE-588)4001170-7 |D s |
| 689 | 1 | 1 | |a Fermatscher Satz |0 (DE-588)4325816-5 |D s |
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| 830 | 0 | |a Graduate texts in mathematics |v 50 |w (DE-604)BV000000067 |9 50 | |
| 856 | 4 | 2 | |m DNB Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008913103&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 883 | 1 | |8 1\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
| 883 | 1 | |8 2\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-008913103 | |
Datensatz im Suchindex
| _version_ | 1807319882109288448 |
|---|---|
| adam_text |
CONTENTS
CHAPTER
1
FERMAT
1
1.1
FERMAT
AND
HIS
"
LAST
THEOREM.
"
STATEMENT
OF
THE
THEOREM.
HISTORY
OF
ITS
DISCOVERY.
1.2
PYTHAGOREAN
TRIANGLES.
PYTHAGOREAN
TRIPLES
KNOWN
TO
THE
BABYLONIANS
1000
YEARS
BEFORE
PYTHAGORAS.
1.3
HOW
TO
FIND
PYTHAGOREAN
TRIPLES.
METHOD
BASED
ON
THE
FACT
THAT
THE
PRODUCT
OF
TWO
RELATIVELY
PRIME
NUMBERS
CAN
BE
A
SQUARE
ONLY
IF
BOTH
FACTORS
ARE
SQUARES.
1.4
THE
METHOD
OF
INFINITE
DESCENT.
1.5
THE
CASE
N
=
4
OF
THE
LAST
THEOREM.
IN
THIS
CASE
THE
PROOF
IS
A
SIMPLE
APPLICATION
OF
INFINITE
DESCENT.
GENERAL
THEOREM
REDUCES
TO
THE
CASE
OF
PRIME
EXPONENTS.
1.6
FERMAT
'
S
ONE
PROOF.
THE
PROOF
THAT
A
PYTHAGOREAN
TRIANGLE
CANNOT
HAVE
AREA
A
SQUARE
INVOLVES
ELEMENTARY
BUT
VERY
INGENIOUS
ARGUMENTS.
1.7
SUMS
OF
TWO
SQUARES
AND
RELATED
TOPICS.
FERMAT
'
S
DISCOVERIES
ABOUT
REPRESENTATIONS
OF
NUMBERS
IN
THE
FORM
N
=
X
2
+
KN
2
FOR
K=
1,2,3.
THE
DIFFERENT
PATTERN
WHEN
K
=
5.
1.8
PERFECT
NUMBERS
AND
FERMAT
'
S
THEOREM.
EUCLID
'
S
FORMULA
FOR
PERFECT
NUMBERS
LEADS
TP
THE
STUDY
OF
MERSENNE
PRIMES
2"
-
1
WHICH
IN
TURN
LEADS
TO
FERMAT
'
S
THEOREM
A
P
-
A
=
0MODP.
PROOF
OF
FERMAT
'
S
THEOREM.
FERMAT
NUMBERS.
THE
FALSE
CONJECTURE
THAT
2
32
+
1
IS
PRIME.
1.9
PELL
'
S
EQUATION.
FERMAT
'
S
CHALLENGE
TO
THE
ENGLISH.
THE
CYCLIC
METHOD
INVENTED
BY
THE
ANCIENT
INDIANS
FOR
THE
SOLUTION
OF
?LX
2
+
1
=Y
2
FOR
GIVEN
NONSQUARE
A.
MISNAMING
OF
THIS
EQUA
TION
AS
"
PELL
'
S
EQUATION
"
BY
EULER.
EXERCISES:
PROOF
THAT
PELL
'
S
EQUATION
ALWAYS
HAS
AN
INFINITY
OF
SOLUTIONS
AND
THAT
THE
CYCLIC
METHOD
PRODUCES
THEM
ALL.
1.10
OTHER
NUMBER-THEORETIC
DISCOVERIES
OF
FERMAT.
FERMAT
'
S
LEGACY
OF
CHALLENGE
PROBLEMS
AND
THE
SOLUTIONS
OF
THESE
PROBLEMS
AT
THE
HANDS
OF
LAGRANGE,
EULER,
GAUSS,
CAUCHY,
AND
OTHERS.
CHAPTER
2
EULER
39
2.1
EULER
AND
THE
CASE
N
=
3.
EULER
NEVER
PUBLISHED
A
CORRECT
PROOF
THAT
X
3
+Y
3
=#Z
3
BUT
THIS
THEOREM
CAN
BE
PROVED
USING
HIS
TECHNIQUES.
2.2
EULER
'
S
PROOF
OF
THE
CASE
N
=
3.
REDUCTION
OF
FERMAT
'
S
LAST
THEOREM
IN
THE
CASE
W
=
3
TO
THE
STATEMENT
THATP
2
+
3^
2
CAN
BE
A
CUBE
(P
AND
Q
RELATIVELY
PRIME)
ONLY
IF
THERE
EXIST
A
AND
B
SUCH
THAT
P
=
A
2
-
9AB
2
,
Q
=
3A
2
B
-
3B
2
.
CONTENTS
2.3
ARITHMETIC
OF
SURDS.
THE
CONDITION
FOR
P
2
+
3 ?
2
TO
BE
A
CUBE
CAN
BE
WRITTEN
SIMPLY
AS
P
+
Q^J
-
3
=
(A
+AV
-
3
)
3
,
THAT
IS,
P
+
/V
-
3
IS
A
CUBE.
EULER
'
S
FALLACIOUS
PROOF,
USING
UNIQUE
FACTORIZATION,
THAT
THIS
CONDI
TION
IS
NECESSARY
FOR
P
2
+
3 ?
2
=
CUBE.
2.4
EULER
ON
SUMS
OF
TWO
SQUARES.
EULER
'
S
PROOFS
OF
THE
BASIC
THEOREMS
CONCERNING
REPRESENTATIONS
OF
NUM
BERS
IN
THE
FORMS
X
2
+Y
2
AND
X
2
+
3Y
2
.
EXERCISES:
NUMBERS
OF
THE
FORM
X
2
+
2Y
2
.
2.5
REMAINDER
OF
THE
PROOF
WHEN
N
=
3.
USE
OF
EULER
'
S
TECHNIQUES
TO
PROVE
X
3
+Y
3
^=Z
3
.
2.6
ADDENDUM
ON
SUMS
OF
TWO
SQUARES.
METHOD
FOR
SOLVING
P
=
X
2
+Y
2
WHEN
P
IS
A
PRIME
OF
THE
FORM
4N
+
1.
SOLVING
P
=
X
2
+
3Y
2
ANDP
=
X
2
+
2Y
2
.
CHAPTER
3
FROM
EULER
TO
KUMMER
59
3.1
INTRODUCTION.
LAGRANGE,
LEGENDRE,
AND
GAUSS.
3.2
SOPHIE
GERMAIN
'
S
THEOREM.
SOPHIE
GERMAIN.
DIVISION
OF
FERMAT
'
S
LAST
THEOREM
INTO
TWO
CASES,
CASE
I
(X,
Y,
AND
Z
RELATIVELY
PRIME
TO
THE
EXPONENT
P)
AND
CASE
II
(OTHERWISE).
SOPHIE
GERMAIN
'
S
THEOREM
IS
A
SUFFICIENT
CONDITION
FOR
CASE
I.
IT
EASILY
PROVES
CASE
I
FOR
ALL
SMALL
PRIMES.
3.3
THE
CASE
N
=
5.
PROOF
THAT
X
5
+Y
5
^Z
5
.
THE
JOINT
ACHIEVEMENT
OF
DIRICHLET
AND
LEGENDRE.
GENERAL
TECHNIQUE
IS
LIKE
EULER
'
S
PROOF
THAT
X
3
+Y
3
=^Z
3
EXCEPT
THAT
P
2
-
5Q
2
A
FIFTH
POWER
IMPLIES
P
+
QV5
=
(A
+
B
V5
)
5
ONLY
UNDER
THE
ADDITIONAL
CONDITION
5
1
7.
3.4
THE
CASES
N
=
14
AND
W
=
7.
THESE
PROOFS,
BY
DIRICHLET
AND
LAME
RESPECTIVELY,
ARE
NOT
EXPLAINED
HERE.
TO
GO
FURTHER
AND
PROVE
FERMAT
'
S
LAST
THEOREM
FOR
LARGER
EXPONENTS
CLEARLY
REQUIRES
NEW
TECHNIQUES.
EX
ERCISE:
DIRICHLET
'
S
PROOF
OF
THE
CASE
N=
14.
CHAPTER
4
KUMMER
'
S
THEORY
OF
IDEAL
FACTORS
76
4.1
THE
EVENTS
OF
1847.
LAME
'
S
"
PROOF
"
OF
FERMAT
'
S
LAST
THEOREM.
LIOUVILLE
'
S
OBJECTION.
CAUCHY
'
S
ATTEMPTS
AT
A
PROOF.
KUMMER
'
S
LETTER
TO
LIOUVILLE.
FAILURE
OF
UNIQUE
FACTORIZATION.
KUMMER
'
S
NEW
THEORY
OF
IDEAL
COMPLEX
NUMBERS.
4.2
CYCLOTOMIC
INTEGERS.
BASIC
DEFINITIONS
AND
OPERA
TIONS.
THE
NORM
OF
A
CYCLOTOMIC
INTEGER.
THE
DISTINCTION
BETWEEN
"
PRIME
"
AND
"IRREDUCIBLE.
"
DIVISION
USING
THE
NORM.
43
FACTORIZATION
OF
PRIMES
P
=
L
MOD
X.
DERIVATION
OF
NECESSARY
AND
SUFFICIENT
CONDITIONS
FOR
A
CYCLOTOMIC
INTEGER
TO
BE
A
PRIME
FACTOR
OF
SUCH
A
PRIME
P.
4.4
COMPUTA
TIONS
WHEN
P=L
MOD
A.
EXPLICIT
FACTORIZATIONS
OF
SUCH
PRIMES
FOR
SMALL
VALUES
OF
P
AND
X.
KUMMER
'
S
FACTORIZATIONS
FOR
X
19
AND
P
1000.
IMPOSSIBILITY
OF
FACTORIZATION
WHEN
X
=
23
AND
P=47.
THE
IDEA
BEHIND
KUMMER
'
S
"
IDEAL
"
PRIME
FACTORS.
4.5
PERIODS.
THE
CONJUGATION
A:A
-
A
7
CORRESPONDING
TO
A
PRIMITIVE
ROOT
Y
MOD
X.
A
CYCLOTOMIC
INTEGER
IS
MADE
UP
OF
PERIODS
OF
LENGTH
/
IF
AND
ONLY
IF
IT
IS
INVARIANT
UNDER
A
C
WHERE
EF=X
-
1.
4.6
FACTORIZATION
OF
PRIMES
P^L
MOD
X.
IF
F
IS
THE
EXPONENT
OF
P
MOD
X
AND
IF
H(A)
IS
ANY
PRIME
FACTOR
OF
P
THEN
THE
PERIODS
OF
LENGTH
F
ARE
ALL
CONGRUENT
TO
INTEGERS
MOD
A
(A).
THIS
MAKES
IT
EASY
TO
TEST
CYCLOTOMIC
INTEGERS
MADE
UP
OF
PERIODS
FOR
DIVISIBILITY
BY
H(A).
4.7
COMPUTATIONS
WHEN
P^L
MOD
X.
EXPLICIT
FACTORIZATIONS
FOR
SMALL
VALUES
OF
P
AND
X.
4.8
EXTENSION
OF
THE
DIVISIBILITY
TEST.
TESTING
ARBITRARY
CYCLOTOMIC
INTEGERS
-
NOT
JUST
THOSE
MADE
UP
OF
PERIODS
-
FOR
DIVISIBILITY
BY
A
GIVEN
PRIME
CYCLOTOMIC
INTEGER
A
(A).
4.9
PRIME
DIVISORS.
THE
TESTS
FOR
DIVISIBILITY
BY
PRIME
FACTORS
EXIST
IN
ALL
CASES,
EVEN
THOSE
IN
WHICH
THERE
IS
NO
PRIME
FACTOR.
THIS
IS
THE
BASIS
FOR
THE
DEFINITION
OF
"
IDEAL
"
PRIME
FACTORS
OR
PRIME
DIVISORS.
INADEQUACY
OF
KUMMER
'
S
ORIGINAL
PROOF
OF
THE
BASIC
PROPOSITION.
4.10
MULTIPLICITIES
AND
THE
EXCEPTIONAL
PRIME.
DEFINITION
OF
THE
MULTIPLICITY
XII
CONTENTS
WITH
WHICH
A
PRIME
DIVISOR
DIVIDES
A
CYCLOTOMIC
INTEGER.
THE
ONE
PRIME
DIVISOR
(1
-
A)
OF
X.
4.11
THE
FUNDAMENTAL
THEOREM.
A
CYCLOTOMIC
INTEGER
G(A)
DIVIDES
ANOTHER
A(A)
IF
AND
ONLY
IF
EVERY
PRIME
DIVISOR
WHICH
DIVIDES
G(A)
DIVIDES
H(A)
WITH
MULTIPLICITY
AT
LEAST
AS
GREAT.
4.12
DIVISORS.
DEFINI
TION
OF
DIVISORS.
NOTATION.
4.13
TERMINOLOGY.
A
DIVISOR
IS
DETERMINED
BY
THE
SET
OF
ALL
THINGS
THAT
IT
DIVIDES.
"
IDEALS.
"
4.14
CONJUGATIONS
AND
THE
NORM
OF
A
DIVISOR.
CONJUGATES
OF
A
DIVISOR.
NORM
OF
A
DIVISOR
AS
A
DIVISOR
AND
AS
AN
INTEGER.
THERE
ARE
N(A)
CLASSES
OF
CYCLOTOMIC
INTEGERS
MOD
A.
THE
CHINESE
REMAINDER
THEOREM.
4.15
SUMMARY.
CHAPTER5
FERMAT
'
S
LAST
THEOREM
FOR
REGULAR
PRIMES
152
5.1
KUMMER
'
S
REMARKS
ON
QUADRATIC
INTEGERS.
THE
NOTION
OF
EQUIVALENCE
OF
DIVISORS.
KUMMER
'
S
ALLUSION
TO
A
THEORY
OF
DIVISORS
FOR
QUADRATIC
INTEGERS
X+YVD
AND
ITS
CONNECTION
WITH
GAUSS
'
S
THEORY
OF
BINARY
QUADRATIC
FORMS.
5.2
EQUIVALENCE
OF
DIVISORS
IN
A
SPECIAL
CASE.
ANALYSIS
OF
THE
QUESTION
"
WHICH
DIVISORS
ARE
DIVISORS
OF
CYCLOTOMIC
INTEGERS?
"
IN
A
SPECIFIC
CASE.
5.3
THE
CLASS
NUMBER.
DEFINITION
AND
BASIC
PROPERTIES
OF
EQUIVALENCE
OF
DIVISORS.
REPRESENTATIVE
SETS.
PROOF
THAT
THE
CLASS
NUMBER
IS
FINITE.
5.4
KUMMER
'
S
TWO
CONDITIONS.
THE
TYPES
OF
ARGUMENTS
USED
TO
PROVE
FERMAT
'
S
LAST
THEOREM
FOR
THE
EXPONENTS
3
AND
5
MOTIVATE
THE
SINGLING
OUT
OF
THE
PRIMES
X
FOR
WHICH
(A)
THE
CLASS
NUMBER
IS
NOT
DIVISIBLE
BY
X
AND
(B)
UNITS
CONGRUENT
TO
INTEGERS
MOD
X
ARE
XTH
POWERS.
SUCH
PRIMES
ARE
CALLED
"
REGULAR.
"
5.5
THE
PROOF
FOR
REGULAR
PRIMES.
KUMMER
'
S
DEDUC
TION
OF
FERMAT
'
S
LAST
THEOREM
FOR
REGULAR
PRIME
EXPONENTS.
FOR
ANY
UNIT
E(A),
THE
UNIT
E(A)/E(A~')
IS
OF
THE
FORM
A
R
.
5.6
QUADRATIC
RECIPROCITY.
KUMMER
'
S
THEORY
LEADS
NOT
ONLY
TO
A
PROOF
OF
THE
FAMOUS
QUADRATIC
RECIPROCITY
LAW
BUT
ALSO
TO
A
DERIVATION
OF
THE
STATEMENT
OF
THE
LAW.
LEGENDRE
SYMBOLS.
THE
SUPPLEMENTARY
LAWS.
CHAPTER
6
DETERMINATION
OF
THE
CLASS
NUMBER
181
6.1
INTRODUCTION.
THE
MAIN
THEOREM
TO
BE
PROVED
IS
KUMMER
'
S
THEOREM
THAT
X
IS
REGULAR
IF
AND
ONLY
IF
IT
DOES
NOT
DIVIDE
THE
NUMERATORS
OF
THE
BERNOULLI
NUMBERS
B
2
,B
4
,.,B
X
_
3
.
6.2
THE
EULER
PRODUCT
FORMULA.
ANALOG
OF
THE
FORMULA
FOR
THE
CASE
OF
CYCLOTOMIC
INTEGERS.
THE
CLASS
NUMBER
FORMULA
IS
FOUND
BY
MULTIPLYING
BOTH
SIDES
BY
(V
-
1)
AND
EVALUATING
THE
LIMIT
AS
S|L.
63
FIRST
STEPS.
PROOF
OF
THE
GENERALIZED
EULER
PRODUCT
FORMULA.
THE
RIEMANN
ZETA
FUNCTION.
6.4
REFORMULATION
OF
THE
RIGHT
SIDE.
THE
RIGHT
SIDE
IS
EQUAL
TO
(
S
)L(
S
,X
I
)L(
S
,
X
2
)'
''
M^XX-2)
WHERE
THE
X
'
S
ARE
THE
NONPRINCIPAL
CHARACTERS
MOD
X.
6.5
DIRICHLET
'
S
EVALUATION
OF
SUMMATION
BY
PARTS.
L(L,X)
AS
A
SUPERPOSITION
OF
THE
SERIES
FOR
LOG(L/(L
-
A
7
)),
J=
1,2,.
,,X
-
1.
EXPLICIT
FORMULAS
FOR
L(L,X)
6.6
THE
LIMIT
OF
THE
RIGHT
SIDE.
AN
EXPLICIT
FORMULA.
6.7
THE
NONVANISHING
OF
L-SERIES.
PROOF
THAT
L(L,X)^0
FOR
THE
X
'
S
UNDER
CONSIDERATION.
6.8
REFOR
MULATION
OF
THE
LEFT
SIDE.
IN
THE
LIMIT
AS
SJL,
THE
SUM
OF
N(A)~
S
OVER
ALL
DIVISORS
A
IN
A
DIVISOR
CLASS
IS
THE
SAME
FOR
ANY
TWO
CLASSES.
PROGRAM
FOR
THE
EVALUATION
OF
THEIR
COMMON
LIMIT.
6.9
UNITS;
THE
FIRST
FEW
CASES.
EXPLICIT
DERIVATION
OF
ALL
UNITS
IN
THE
CASES
X
=
3,5,7.
FINITE-DIMENSIONAL
FOURIER
ANALYSIS.
IMPLICIT
DERIVATION
OF
THE
UNITS
IN
THE
CASE
X
=
11.
SECOND
FACTOR
OF
THE
CLASS
NUMBER.
6.10
UNITS:
THE
GENERAL
CASE.
METHOD
FOR
FINDING,
AT
LEAST
IN
PRINCIPLE,
ALL
UNITS.
SUM
OVER
ALL
PRINCIPAL
DIVISORS
WRITTEN
IN
TERMS
OF
A
SUM
OVER
A
CERTAIN
SET
OF
CYCLOTOMIC
INTEGERS.
6.11
EVALUATION
OF
THE
INTEGRAL.
SOLUTION
OF
A
PROBLEM
IN
INTEGRAL
CALCULUS.
6.12
XIII
CONTENTS
COMPARISON
OF
THE
INTEGRAL
AND
THE
SUM.
IN
THE
LIMIT
TO
BE
EVALUATED,
THE
SUM
CAN
BE
REPLACED
BY
THE
INTEGRAL.
6.13
THE
SUM
OVER
OTHER
DIVISOR
CLASSES.
PROOF
THAT,
IN
THE
LIMIT,
THE
SUM
OVER
ANY
TWO
DIVISOR
CLASSES
IS
THE
SAME.
6.14
THE
CLASS
NUMBER
FORMULA.
ASSEMBLING
OF
ALL
THE
PIECES
OF
THE
PRECEDING
SECTIONS
TO
GIVE
THE
EXPLICIT
FORMULA
FOR
THE
CLASS
NUMBER.
6.15
PROOF
THAT
37
IS
IRREGULAR.
SIMPLIFICATIONS
OF
THE
COMPUTATION
OF
THE
FIRST
FACTOR
OF
THE
CLASS
NUMBER.
BERNOULLI
NUMBERS
AND
BERNOULLI
POLYNOMIALS.
6.16
DIVISIBILITY
OF
THE
FIRST
FACTOR
BY
A.
GENERALIZATION
OF
THE
TECHNIQUES
OF
THE
PRECEDING
SECTION
TO
SHOW
THAT
X
DIVIDES
THE
FIRST
FACTOR
OF
THE
CLASS
NUMBER
IF
AND
ONLY
IF
IT
DIVIDES
THE
NUMERATOR
OF
ONE
OF
THE
BERNOULLI
NUMBERS
B
2
,B
4
,.,B
A
_
3
.
6.17
DIVISIBILITY
OF
THE
SECOND
FACTOR
BY
A.
PROOF
THAT
X
DIVIDES
THE
SECOND
FACTOR
OF
THE
CLASS
NUMBER
ONLY
IF
IT
ALSO
DIVIDES
THE
FIRST
FACTOR.
6.18
KUMMER
'
S
LEMMA.
(A)
IMPLIES
(B).
6.19
SUMMARY.
CHAPTER
7
DIVISOR
THEORY
FOR
QUADRATIC
INTEGERS
245
7.1
THE
PRIME
DIVISORS.
DETERMINATION
OF
WHAT
THE
PRIME
DIVISORS
MUST
BE
IF
THERE
IS
TO
BE
A
DIVISOR
THEORY
FOR
NUMBERS
OF
THE
FORM
X+Y\FD
.
MODIFICATION
OF
THE
DEFINITION
OF
QUADRATIC
INTEGERS
IN
THE
CASE
D
=
LMOD4.
7.2
THE
DIVISOR
THEORY.
PROOF
THAT
THE
DIVISORS
DEFINED
IN
THE
PRECEDING
SECTION
GIVE
A
DIVISOR
THEORY
WITH
ALL
THE
EXPECTED
PROPERTIES.
EQUIVALENCE
OF
DIVISORS.
73
THE
SIGN
OF
THE
NORM.
WHEN
D
0
THE
NORM
ASSUMES
NEGATIVE
AS
WELL
AS
POSITIVE
VALUES.
IN
THIS
CASE
A
DIVISOR
WITH
NORM
-
1
IS
INTRODUCED.
7.4
QUADRATIC
INTEGERS
WITH
GIVEN
DIVISORS.
UNLIKE
THE
CYCLOTOMIC
CASE,
FOR
QUADRATIC
INTEGERS
THERE
IS
A
SIMPLE
ALGORITHM
FOR
DETERMINING
WHETHER
A
GIVEN
DIVISOR
IS
PRINCIPAL
AND,
IF
SO,
OF
FINDING
ALL
QUADRATIC
INTEGERS
WITH
THIS
DIVISOR.
IT
IS,
IN
ESSENCE,
THE
CYCLIC
METHOD
OF
THE
ANCIENT
INDIANS.
PROOF
OF
THE
VALIDITY
OF
THE
ALGORITHM
IN
THE
CASE
D 0.
EXERCISES:
USE
OF
2X2
MATRICES
TO
STREAMLINE
THE
COMPUTATIONS
OF
THE
CYCLIC
METHOD.
7.5
VALIDITY
OF
THE
CYCLIC
METHOD.
PROOF
IN
THE
CASE
D
0.
COMPUTATION
OF
THE
FUNDAMENTAL
UNIT.
7.6
THE
DIVISOR
CLASS
GROUP:
EXAMPLES.
EXPLICIT
DERIVATION
OF
THE
DIVISOR
CLASS
GROUP
FOR
SEVERAL
VALUES
OF
D.
7.7
THE
DIVISOR
CLASS
GROUP:
A
GENERAL
THEOREM.
PROOF
THAT
TWO
DIVISORS
ARE
EQUIVALENT
ONLY
IF
APPLICATION
OF
THE
CYCLIC
METHOD
TO
THEM
YIELDS
THE
SAME
PERIOD
OF
REDUCED
DIVISORS.
THIS
SIMPLIFIES
THE
DERIVATION
OF
THE
DIVISOR
CLASS
GROUP.
7.8
EULER
'
S
THEOREMS.
EULER
FOUND
EMPIRICALLY
THAT
THE
WAY
IN
WHICH
A
PRIME
P
FACTORS
IN
QUADRATIC
INTEGERS
X+Y\Z~D
DEPENDS
ONLY
ON
THE
CLASS
OF
P
MOD
4Z .
HE
FOUND
OTHER
THEOREMS
WHICH
SIMPLIFY
THE
DETERMINATION
OF
THE
CLASSES
OF
PRIMES
MOD
4Z)
WHICH
SPLIT
AND
THE
CLASSES
WHICH
REMAIN
PRIME.
THESE
THEOREMS,
UNPROVED
BY
EULER,
IMPLY
AND
ARE
IMPLIED
BY
THE
LAW
OF
QUADRATIC
RECIPROCITY.
7.9
GENERA.
GAUSS
'
S
NECESSARY
CONDITIONS
FOR
TWO
DIVISORS
TO
BE
EQUIVALENT.
CHARACTER
OF
A
DIVISOR
CLASS.
RESULTING
PARTITION
OF
THE
DIVISOR
CLASSES
INTO
GENERA.
7.10
AMBIGUOUS
CLASSES.
DEFINITION.
PROOF
THAT
THE
NUMBER
OF
AMBIGUOUS
CLASSES
IS
AT
MOST
HALF
THE
NUMBER
OF
POSSIBLE
CHARACTERS.
7.11
GAUSS
'
S
SECOND
PROOF
OF
QUADRATIC
RECIPROCITY.
PROOF
THAT
AT
MOST
HALF
OF
THE
POSSIBLE
CHARACTERS
ACTUALLY
OCCUR.
GAUSS
'
S
DEDUCTION,
FROM
THIS
THEOREM,
OF
QUADRATIC
RECIPROCITY.
CHAPTER
8
GAUSS
'
S
THEORY
OF
BINARY
QUADRATIC
FORMS
305
8.1
OTHER
DIVISOR
CLASS
GROUPS.
WHEN
D
IS
NOT
SQUAREFREE
THE
DEFINITION
OF
THE
DIVISOR
CLASS
GROUP
NEEDS
TO
BE
MODIFIED.
ORDERS
OF
QUADRATIC
INTEGERS.
XIV
CONTENTS
EQUIVALENCE
RELATIVE
TO
AN
ORDER.
THE
DIVISOR
CLASS
GROUP
CORRESPONDING
TO
AN
ORDER.
EXERCISES:
EULER
'
S
CONVENIENT
NUMBERS.
8.2
ALTERNATIVE
VIEW
OF
THE
CYCLIC
METHOD.
INTERPRETATION
OF
IT
AS
A
METHOD
FOR
GENERATING
EQUIV
ALENT
BINARY
QUADRATIC
FORMS.
METHOD
FOR
FINDING
REPRESENTATIONS
OF
GIVEN
INTEGERS
BY
GIVEN
BINARY
QUADRATIC
FORMS.
8.3
THE
CORRESPONDENCE
BETWEEN
DIVISORS
AND
BINARY
QUADRATIC
FORMS.
PROPER
EQUIVALENCE
OF
BINARY
QUADRATIC
FORMS.
THE
ONE-TO-ONE
CORRESPONDENCE
BETWEEN
PROPER
EQUIVA
LENCE
CLASSES
OF
PROPERLY
PRIMITIVE
FORMS
(POSITIVE
WHEN
D
0)
AND
DIVISOR
CLASSES
FOR
THE
ORDER
{X+YVD
:
X,Y
INTEGERS}.
8.4
THE
CLASSIFICA
TION
OF
FORMS.
EXTENSION
OF
THE
THEOREM
OF
SECTION
7.7
TO
THE
CASE
WHERE
D
IS
NOT
SQUAREFREE.
8.5
EXAMPLES.
DERIVATION
OF
THE
DIVISOR
CLASS
GROUP
IN
SEVERAL
CASES.
8.6
GAUSS
'
S
COMPOSITION
OF
FORMS.
HOW
GAUSS
DEFINED
THE
PRODUCT
OF
TWO
CLASSES
OF
BINARY
QUADRATIC
FORMS
WITHOUT
USING
DIVISOR
THEORY.
8.7
EQUATIONS
OF
DEGREE
2
IN
2
VARIABLES.
COMPLETE
SOLUTION,
ESSENTIALLY
DUE
TO
LAGRANGE,
OF
AX
2
+
BXY
+
CY
2
+
DX
+
EY
+F
=
0.
CHAPTER
9
DIRICHLET
'
S
CLASS
NUMBER
FORMULA
342
9.1
THE
EULER
PRODUCT
FORMULA.
ANALOG
IN
THE
CASE
OF
QUADRATIC
INTEGERS.
SPLITTING
INTO
CASES
FOR
VARIOUS
TYPES
OF
D.
9.2
FIRST
CASE.
THE
CASE
D
0,
7)^1
MOD
4,
D
SQUAREFREE.
DERIVATION
OF
THE
CLASS
NUMBER
FORMULA.
EXAMPLES.
9.3
ANOTHER
CASE.
THE
CASE
D
0,
LMOD4,
D
SQUAREFREE.
DERIVATION.
EXAMPLES.
9.4
D
=LMOD4.
MODIFICATIONS
REQUIRED
WHEN
D
=
LMOD4,
D
SQUAREFREE.
9.5
EVALUATION
OF
S()-.
THIS
TERM
OF
THE
CLASS
NUMBER
FORMULA
CAN
BE
EVALUATED
USING
THE
TECHNIQUE
OF
SECTION
6.5.
FOURIER
TRANSFORM
OF
THE
CHARACTER
()
MOD
4Z
IS
A
MULTIPLE
OF
ITSELF.
USE
OF
THIS
FACT
TO
REDUCE
THE
FORMULA.
EXERCISES:
DIRICHLET
'
S
FURTHER
REDUCTIONS
OF
THE
FORMULA
IN
THE
CASE
D
0,
D
SQUAREFREE.
THE
SIGN
OF
GAUSSIAN
SUMS
AND
ITS
RELATION
TO
THIS
FORMULA.
9.6
SUBORDERS.
GENERALIZATION
OF
THE
CLASS
NUMBER
FORMULA
TO
THE
CASE
WHERE
D
IS
NOT
SQUAREFREE
AND,
MORE
GENER
ALLY,
TO
DIVISOR
CLASS
GROUPS
CORRESPONDING
TO
ARBITRARY
ORDERS
OF
QUADRATIC
INTEGERS.
9.7
PRIMES
IN
ARITHMETIC
PROGRESSIONS.
DIRICHLET
'
S
PROOF
THAT
AN
+
B
REPRESENTS
AN
INFINITY
OF
PRIMES
WHEN
B
IS
RELATIVELY
PRIME
TO
A.
USE
OF
THE
CLASS
NUMBER
FORMULA
TO
PROVE
L(L,X)
:
#0
FOR
ALL
REAL
CHARACTERS
X
MOD
A.
APPENDIX:
THE
NATURAL
NUMBERS
372
A.L
BASIC
PROPERTIES.
ADDITION
AND
MULTIPLICATION.
EUCLIDEAN
ALGORITHM.
CONGRUENCE
MODULO
A
NATURAL
NUMBER.
CHINESE
REMAINDER
THEOREM.
SOLU
TION
OF
AX
=
B
MODE.
FUNDAMENTAL
THEOREM
OF
ARITHMETIC.
INTEGERS.
A.2
PRIMITIVE
ROOTS
MOD
P.
DEFINITION.
PROOF
THAT
EVERY
P
HAS
A
PRIMITIVE
ROOT.
ANSWERS
TO
EXERCISES
BIBLIOGRAPHY
INDEX
381
403
409
XV |
| any_adam_object | 1 |
| author | Edwards, Harold M. 1936-2020 |
| author_GND | (DE-588)117709700 |
| author_facet | Edwards, Harold M. 1936-2020 |
| author_role | aut |
| author_sort | Edwards, Harold M. 1936-2020 |
| author_variant | h m e hm hme |
| building | Verbundindex |
| bvnumber | BV013084054 |
| callnumber-first | Q - Science |
| callnumber-label | QA247 |
| callnumber-raw | QA247 |
| callnumber-search | QA247 |
| callnumber-sort | QA 3247 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 180 |
| classification_tum | MAT 120f |
| ctrlnum | (OCoLC)44119521 (DE-599)BVBBV013084054 |
| dewey-full | 512.74 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.74 |
| dewey-search | 512.74 |
| dewey-sort | 3512.74 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. softcover printing |
| format | Book |
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| id | DE-604.BV013084054 |
| illustrated | Not Illustrated |
| indexdate | 2024-08-14T00:15:51Z |
| institution | BVB |
| isbn | 0387950028 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008913103 |
| oclc_num | 44119521 |
| open_access_boolean | |
| owner | DE-703 DE-91G DE-BY-TUM |
| owner_facet | DE-703 DE-91G DE-BY-TUM |
| physical | XV, 410 S. 25 cm |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Edwards, Harold M. 1936-2020 Verfasser (DE-588)117709700 aut Fermat's last theorem a genetic introduction to algebraic number theory Harold M. Edwards 1. softcover printing New York [u.a.] Springer 2000 XV, 410 S. 25 cm txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 50 Literaturverz. S. 403 - 407 Algebraic number theory Fermat's last theorem Algebraische Zahlentheorie (DE-588)4001170-7 gnd rswk-swf Fermatsche Vermutung (DE-588)4154012-8 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Fermatscher Satz (DE-588)4325816-5 gnd rswk-swf Fermatsche Vermutung (DE-588)4154012-8 s Algebraische Zahlentheorie (DE-588)4001170-7 s DE-604 Fermatscher Satz (DE-588)4325816-5 s 1\p DE-604 Zahlentheorie (DE-588)4067277-3 s 2\p DE-604 Graduate texts in mathematics 50 (DE-604)BV000000067 50 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008913103&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Edwards, Harold M. 1936-2020 Fermat's last theorem a genetic introduction to algebraic number theory Graduate texts in mathematics Algebraic number theory Fermat's last theorem Algebraische Zahlentheorie (DE-588)4001170-7 gnd Fermatsche Vermutung (DE-588)4154012-8 gnd Zahlentheorie (DE-588)4067277-3 gnd Fermatscher Satz (DE-588)4325816-5 gnd |
| subject_GND | (DE-588)4001170-7 (DE-588)4154012-8 (DE-588)4067277-3 (DE-588)4325816-5 |
| title | Fermat's last theorem a genetic introduction to algebraic number theory |
| title_auth | Fermat's last theorem a genetic introduction to algebraic number theory |
| title_exact_search | Fermat's last theorem a genetic introduction to algebraic number theory |
| title_full | Fermat's last theorem a genetic introduction to algebraic number theory Harold M. Edwards |
| title_fullStr | Fermat's last theorem a genetic introduction to algebraic number theory Harold M. Edwards |
| title_full_unstemmed | Fermat's last theorem a genetic introduction to algebraic number theory Harold M. Edwards |
| title_short | Fermat's last theorem |
| title_sort | fermat s last theorem a genetic introduction to algebraic number theory |
| title_sub | a genetic introduction to algebraic number theory |
| topic | Algebraic number theory Fermat's last theorem Algebraische Zahlentheorie (DE-588)4001170-7 gnd Fermatsche Vermutung (DE-588)4154012-8 gnd Zahlentheorie (DE-588)4067277-3 gnd Fermatscher Satz (DE-588)4325816-5 gnd |
| topic_facet | Algebraic number theory Fermat's last theorem Algebraische Zahlentheorie Fermatsche Vermutung Zahlentheorie Fermatscher Satz |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008913103&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT edwardsharoldm fermatslasttheoremageneticintroductiontoalgebraicnumbertheory |