Galois Theory: Graduate Texts in Mathematics, cartea 204
Traducere de L. Schneps Autor Jean-Pierre Escofieren Limba Engleză Hardback – 20 dec 2000
Acest manual academic, publicat sub egida prestigioasei serii Graduate Texts in Mathematics de la Springer, reprezintă o introducere riguroasă, dar accesibilă în fundamentele teoriei lui Galois. Autorul Jean-Pierre Escofier adoptă o abordare pedagogică distinctă, integrând organic un bogat material istoric în expunerea matematică. Notăm cu interes că volumul nu se limitează la abstractizarea modernă, ci reconstruiește contextul rezolvării ecuațiilor algebrice, pornind de la metodele geometrice ale grecilor și algebra arabă, până la soluțiile radicale ale lui Cardan și Tartaglia. Reținem ca element definitoriu prezența unui set copios de exerciții selectate cu atenție, care nu sunt simple anexe, ci piloni ai procesului de învățare, facilitând tranziția de la calculul concret la structurile de grup și corp. Ca alternativă la Galois Theory Through Exercises de Juliusz Brzezinski pentru cursurile de algebră abstractă, lucrarea lui Escofier are avantajul unei perspective istorice mai vaste și al unui ritm de prezentare mai așezat, fiind ideală pentru studenții care preferă să înțeleagă geneza conceptelor înainte de a le formaliza complet. Structura este clară: primele capitole sunt dedicate evoluției rezolvării ecuațiilor și polinoamelor simetrice, pregătind terenul pentru nucleul teoriei lui Galois, în timp ce exemplele computaționale și dezvoltările recente ancorează materia în cercetarea contemporană.
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Specificații
ISBN-13: 9780387987651
ISBN-10: 0387987657
Pagini: 283
Ilustrații: 1
Dimensiuni: 155 x 235 x 23 mm
Greutate: 0.56 kg
Ediția:2001
Editura: Springer
Colecția Springer
Seria Graduate Texts in Mathematics
Locul publicării:New York, NY, United States
ISBN-10: 0387987657
Pagini: 283
Ilustrații: 1
Dimensiuni: 155 x 235 x 23 mm
Greutate: 0.56 kg
Ediția:2001
Editura: Springer
Colecția Springer
Seria Graduate Texts in Mathematics
Locul publicării:New York, NY, United States
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Public țintă
GraduateDe ce să citești această carte
Este o resursă esențială pentru studenții la matematică (nivel masterat) care doresc să stăpânească teoria lui Galois fără a pierde din vedere contextul istoric. Cititorul câștigă o înțelegere profundă a modului în care s-au cristalizat conceptele moderne de algebră, beneficiind de numeroase exerciții rezolvate care clarifică pașii tehnici dificili ai demonstrațiilor clasice.
Descriere scurtă
This
book
offers
the
fundamentals
of
Galois
Theory,
including
a
set
of
copious,
well-chosen
exercises
that
form
an
important
part
of
the
presentation.
The
pace
is
gentle
and
incorporates
interesting
historical
material,
including
aspects
on
the
life
of
Galois.
Computed
examples,
recent
developments,
and
extensions
of
results
into
other
related
areas
round
out
the
presentation.
Cuprins
1
Historical
Aspects
of
the
Resolution
of
Algebraic
Equations.-
1.1
Approximating
the
Roots
of
an
Equation.-
1.2
Construction
of
Solutions
by
Intersections
of
Curves.-
1.3
Relations
with
Trigonometry.-
1.4
Problems
of
Notation
and
Terminology.-
1.5
The
Problem
of
Localization
of
the
Roots.-
1.6
The
Problem
of
the
Existence
of
Roots.-
1.7
The
Problem
of
Algebraic
Solutions
of
Equations.-
Toward
Chapter
2.-
2
Resolution
of
Quadratic,
Cubic,
and
Quartic
Equations.-
2.1
Second-Degree
Equations.-
2.1.1
The
Babylonians.-
2.1.2
The
Greeks.-
2.1.3
The
Arabs.-
2.1.4
Use
of
Negative
Numbers.-
2.2
Cubic
Equations.-
2.2.1
The
Greeks.-
2.2.2
Omar
Khayyam
and
Sharaf
ad
Din
at
Tusi.-
2.2.3
Scipio
del
Ferro,
Tartaglia,
Cardan.-
2.2.4
Algebraic
Solution
of
the
Cubic
Equation.-
2.2.5
First
Computations
with
Complex
Numbers.-
2.2.6
Raffaele
Bombelli.-
2.2.7
François
Viète.-
2.3
Quartic
Equations.-
Exercises
for
Chapter
2.-
Solutions
to
Some
of
the
Exercises.-
3
Symmetric
Polynomials.-
3.1
Symmetric
Polynomials.-
3.1.1
Background.-
3.1.2
Definitions.-
3.2
Elementary
Symmetric
Polynomials.-
3.2.1
Definition.-
3.2.2
The
Product
of
the
X
?
Xi;
Relations
Between
Coefficients
and
Roots.-
3.3
Symmetric
Polynomials
and
Elementary
Symmetric
Polynomials.-
3.3.1
Theorem.-
3.3.2
Proposition.-
3.3.3
Proposition.-
3.4
Newton’s
Formulas.-
3.5
Resultant
of
Two
Polynomials.-
3.5.1
Definition.-
3.5.2
Proposition.-
3.6
Discriminant
of
a
Polynomial.-
3.6.1
Definition.-
3.6.2
Proposition.-
3.6.3
Formulas.-
3.6.4
Polynomials
with
Real
Coefficients:
Real
Roots
and
Sign
of
the
Discriminant.-
Exercises
for
Chapter
3.-
Solutions
to
Some
of
the
Exercises.-
4
Field
Extensions.-
4.1
Field
Extensions.-
4.1.1
Definition.-
4.1.2
Proposition.-
4.1.3
The
Degree
of
an
Extension.-
4.1.4
Towers
of
Fields.-
4.2
The
Tower
Rule.-
4.2.1
Proposition.-
4.3
Generated
Extensions.-
4.3.1
Proposition.-
4.3.2
Definition.-
4.3.3
Proposition.-
4.4
Algebraic
Elements.-
4.4.1
Definition.-
4.4.2
Transcendental
Numbers.-
4.4.3
Minimal
Polynomial
of
an
Algebraic
Element.-
4.4.4
Definition.-
4.4.5
Properties
of
the
Minimal
Polynomial.-
4.4.6
Proving
the
Irreducibility
of
a
Polynomial
in
Z[X].-
4.5
Algebraic
Extensions.-
4.5.1
Extensions
Generated
by
an
Algebraic
Element.-
4.5.2
Properties
of
K[a].-
4.5.3
Definition.-
4.5.4
Extensions
of
Finite
Degree.-
4.5.5
Corollary:
Towers
of
Algebraic
Extensions.-
4.6
Algebraic
Extensions
Generated
by
n
Elements.-
4.6.1
Notation.-
4.6.2
Proposition.-
4.6.3
Corollary.-
4.7
Construction
of
an
Extension
by
Adjoining
a
Root.-
4.7.1
Definition.-
4.7.2
Proposition.-
4.7.3
Corollary.-
4.7.4
Universal
Property
of
K[X]/(P).-
Toward
Chapters
5
and
6.-
Exercises
for
Chapter
4.-
Solutions
to
Some
of
the
Exercises.-
5
Constructions
with
Straightedge
and
Compass.-
5.1
Constructible
Points.-
5.2
Examples
of
Classical
Constructions.-
5.2.1
Projection
of
a
Point
onto
a
Line.-
5.2.2
Construction
of
an
Orthonormal
Basis
from
Two
Points.-
5.2.3
Construction
of
a
Line
Parallel
to
a
Given
Line
Passing
Through
a
Point.-
5.3
Lemma.-
5.4
Coordinates
of
Points
Constructible
in
One
Step.-
5.5
A
Necessary
Condition
for
Constructibility.-
5.6
Two
Problems
More
Than
Two
Thousand
Years
Old.-
5.6.1
Duplication
of
the
Cube.-
5.6.2
Trisection
of
the
Angle.-
5.7
A
Sufficient
Condition
for
Constructibility.-
Exercises
for
Chapter
5.-
Solutions
to
Some
of
the
Exercises.-
6
K-Homomorphisms.-
6.1
Conjugate
Numbers.-
6.2
K-Homomorphisms.-
6.2.1
Definitions.-
6.2.2
Properties.-
6.3
Algebraic
Elements
and
K-Homomorphisms.-
6.3.1
Proposition.-
6.3.2
Example.-
6.4
Extensions
of
Embeddings
into
?.-
6.4.1
Definition.-
6.4.2
Proposition.-
6.4.3
Proposition.-
6.5
The
Primitive
Element
Theorem.-
6.5.1
Theorem
and
Definition.-
6.5.2
Example.-
6.6
Linear
Independence
of
K-Homomorphisms.-
6.6.1
Characters.-
6.6.2
Emil
Artin’s
Theorem.-
6.6.3
Corollary:
Dedekind’s
Theorem.-
Exercises
for
Chapter
6.-
Solutions
to
Some
of
the
Exercises.-
7
Normal
Extensions.-
7.1
Splitting
Fields.-
7.1.1
Definition.-
7.1.2
Splitting
Field
of
a
Cubic
Polynomial.-
7.2
Normal
Extensions.-
7.3
Normal
Extensions
and
K-Homomorphisms.-
7.4
Splitting
Fields
and
Normal
Extensions.-
7.4.1
Proposition.-
7.4.2
Converse.-
7.5
Normal
Extensions
and
Intermediate
Extensions.-
7.6
Normal
Closure.-
7.6.1
Definition.-
7.6.2
Proposition.-
7.6.3
Proposition.-
7.7
Splitting
Fields:
General
Case.-
Toward
Chapter
8.-
Exercises
for
Chapter
7.-
Solutions
to
Some
of
the
Exercises.-
8
Galois
Groups.-
8.1
Galois
Groups.-
8.1.1
The
Galois
Group
of
an
Extension.-
8.1.2
The
Order
of
the
Galois
Group
of
a
Normal
Extension
of
Finite
Degree.-
8.1.3
The
Galois
Group
of
a
Polynomial.-
8.1.4
The
Galois
Group
as
a
Subgroup
of
a
Permutation
Group.-
8.1.5
A
Short
History
of
Groups.-
8.2
Fields
of
Invariants.-
8.2.1
Definition
and
Proposition.-
8.2.2
Emil
Artin’s
Theorem.-
8.3
The
Example
of
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:
First
Part.-
8.4
Galois
Groups
and
Intermediate
Extensions.-
8.5
The
Galois
Correspondence.-
8.6
The
Example
of
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Second
Part.-
8.7
The
Example
X4
+
2.-
8.7.1
Dihedral
Groups.-
8.7.2
The
Special
Case
of
D4.-
8.7.3
The
Galois
Group
of
X4
+
2.-
8.7.4
The
Galois
Correspondence.-
8.7.5
Search
for
Minimal
Polynomials.-
Toward
Chapters
9,
10,
and
12.-
Exercises
for
Chapter
8.-
Solutions
to
Some
of
the
Exercises.-
9
Roots
of
Unity.-
9.1
The
Group
U(n)
of
Units
of
the
Ring
?/n?.-
9.1.1
Definition
and
Background.-
9.1.2
The
Structure
of
U(n).-
9.2
The
Möbius
Function.-
9.2.1
Multiplicative
Functions.-
9.2.2
The
Möbius
Function.-
9.2.3
Proposition.-
9.2.4
The
Möbius
Inversion
Formula.-
9.3
Roots
of
Unity.-
9.3.1
n-th
Roots
of
Unity.-
9.3.2
Proposition.-
9.3.3
Primitive
Roots.-
9.3.4
Properties
of
Primitive
Roots.-
9.4
Cyclotomic
Polynomials.-
9.4.1
Definition.-
9.4.2
Properties
of
the
Cyclotomic
Polynomial.-
9.5
The
Galois
Group
over
Q
of
an
Extension
of
Q
by
a
Root
of
Unity.-
Exercises
for
Chapter
9.-
Solutions
to
Some
of
the
Exercises.-
10
Cyclic
Extensions.-
10.1
Cyclic
and
Abelian
Extensions.-
10.2
Extensions
by
a
Root
and
Cyclic
Extensions.-
10.3
Irreducibility
of
Xp
?
a.-
10.4
Hilbert’s
Theorem
90.-
10.4.1
The
Norm.-
10.4.2
Hilbert’s
Theorem
90.-
10.5
Extensions
by
a
Root
and
Cyclic
Extensions:
Converse.-
10.6
Lagrange
Resolvents.-
10.6.1
Definition.-
10.6.2
Properties.-
10.7
Resolution
of
the
Cubic
Equation.-
10.8
Solution
of
the
Quartic
Equation.-
10.9
Historical
Commentary.-
Exercises
for
Chapter
10.-
Solutions
to
Some
of
the
Exercises.-
11
Solvable
Groups.-
11.1
First
Definition.-
11.2
Derived
or
Commutator
Subgroup.-
11.3
Second
Definition
of
Solvability.-
11.4
Examples
of
Solvable
Groups.-
11.5
Third
Definition.-
11.6
The
Group
An
Is
Simple
for
n
?
5.-
11.6.1
Theorem.-
11.6.2
An
Is
Not
Solvable
for
n
?
5,
Direct
Proof.-
11.7
Recent
Results.-
Exercises
for
Chapter
11.-
Solutions
to
Some
of
the
Exercises.-
12
Solvability
of
Equations
by
Radicals.-
12.1
Radical
Extensions
and
Polynomials
Solvable
by
Radicals.-
12.1.1
Radical
Extensions.-
12.1.2
Polynomials
Solvable
by
Radicals.-
12.1.3
First
Construction.-
12.1.4
Second
Construction.-
12.2
If
a
Polynomial
Is
Solvable
by
Radicals,
Its
Galois
Group
Is
Solvable.-
12.3
Example
of
a
Polynomial
Not
Solvable
by
Radicals.-
12.4
The
Converse
of
the
Fundamental
Criterion.-
12.5
The
General
Equation
of
Degree
n.-
12.5.1
Algebraically
Independent
Elements.-
12.5.2
Existence
of
Algebraically
Independent
Elements.-
12.5.3
The
General
Equation
of
Degree
n.-
12.5.4
Galois
Group
of
the
General
Equation
of
Degree
n.-
Exercises
for
Chapter
12.-
Solutions
to
Some
of
the
Exercises.-
13
The
Life
of
Évariste
Galois.-
14
Finite
Fields.-
14.1
Algebraically
Closed
Fields.-
14.1.1
Definition.-
14.1.2
Algebraic
Closures.-
14.1.3
Theorem
(Steinitz,
1910).-
14.2
Examples
of
Finite
Fields.-
14.3
The
Characteristic
of
a
Field.-
14.3.1
Definition.-
14.3.2
Properties.-
14.4
Properties
of
Finite
Fields.-
14.4.1
Proposition.-
14.4.2
The
Frobenius
Homomorphism.-
14.5
Existence
and
Uniqueness
of
a
Finite
Field
with
pr
Elements.-
14.5.1
Proposition.-
14.5.2
Corollary.-
14.6
Extensions
of
Finite
Fields.-
14.7
Normality
of
a
Finite
Extension
of
Finite
Fields.-
14.8
The
Galois
Group
of
a
Finite
Extension
of
a
Finite
Field.-
14.8.1
Proposition.-
14.8.2
The
Galois
Correspondence.-
14.8.3
Example.-
Exercises
for
Chapter
14.-
Solutions
to
Some
of
the
Exercises.-
15
Separable
Extensions.-
15.1
Separability.-
15.2
Example
of
an
Inseparable
Element.-
15.3
A
Criterion
for
Separability.-
15.4
Perfect
Fields.-
15.5
Perfect
Fields
and
Separable
Extensions.-
15.6
Galois
Extensions.-
15.6.1
Definition.-
15.6.2
Proposition.-
15.6.3
The
Galois
Correspondence.-
Toward
Chapter
16.-
16
Recent
Developments.-
16.1
The
Inverse
Problem
of
Galois
Theory.-
16.1.1
The
Problem.-
16.1.2
The
Abelian
Case.-
16.1.3
Example.-
16.2
Computation
of
Galois
Groups
over
?
for
Small-Degree
Polynomials.-
16.2.1
Simplification
of
the
Problem.-
16.2.2
The
Irreducibility
Problem.-
16.2.3
Embedding
of
G
into
Sn.-
16.2.4
Looking
for
G
Among
the
Transitive
Subgroups
of
Sn.-
16.2.5
Transitive
Subgroups
of
S4.-
16.2.6
Study
of
?(G)
?
An.-
16.2.7
Study
of
?(G)
?
D4.-
16.2.8
Study
of
?(G)
?
?/4?.-
16.2.9
An
Algorithm
for
n
=
4.
Recenzii
J.-P.
Escofier
Galois Theory
"Escofier’s treatment, at a level suitable for advanced, senior undergraduates or first-year graduate students, centers on finite extensions of number fields, incorporating numerous examples and leaving aside finite fields and the entire concept of separability for the final chapters . . . copious, well-chosen exercises . . . are presented with their solutions . . . The prose is . . . spare and enthusiastic, and the proofs are both instructive and efficient . . . Escofier has written an excellent text, offering a relatively elementary introduction to a beautiful subject in a book sufficiently broad to present a contemporary viewpoint and intuition but sufficiently restrained so as not to overwhelm the reader."—AMERICAN MATHEMATICAL SOCIETY
Galois Theory
"Escofier’s treatment, at a level suitable for advanced, senior undergraduates or first-year graduate students, centers on finite extensions of number fields, incorporating numerous examples and leaving aside finite fields and the entire concept of separability for the final chapters . . . copious, well-chosen exercises . . . are presented with their solutions . . . The prose is . . . spare and enthusiastic, and the proofs are both instructive and efficient . . . Escofier has written an excellent text, offering a relatively elementary introduction to a beautiful subject in a book sufficiently broad to present a contemporary viewpoint and intuition but sufficiently restrained so as not to overwhelm the reader."—AMERICAN MATHEMATICAL SOCIETY