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Fine B., Rosenberger G. — Fundamental Theorem of Algebra
Fine B., Rosenberger G. — Fundamental Theorem of Algebra

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Название: Fundamental Theorem of Algebra

Авторы: Fine B., Rosenberger G.

Аннотация:

The Fundamental Theorem of Algebra states that any complex polynomial must have a complex root. This basic result, whose first accepted proof was given by Gauss, lies really at the intersection of the theory of numbers and the theory of equations and arises also in many other areas of mathematics. The purpose of this book is to examine three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis, and topology. The first proof in each pair is fairly straightforward and depends only on what could be considered elementary mathematics. However, each of these first proofs lends itself to generalizations which in turn lead to more general results from which the fundamental theorem can be deduced as a direct consequence. These general results constitute the second proof in each pair. To arrive at each of the proofs, enough of the general theory of each relevant area is developed to understand the proof. In addition to the proofs and techniques themselves, many applications, such as the insolvability of the quintic and the transcendence of [rho] and [pi] are presented. Finally, a series of appendices give six additional proofs including a version of Gauss's original first proof. The book is intended for junior/senior-level undergraduate mathematics students or first-year graduate students. It is ideal for a "capstone" course in mathematics. It could also be used as an alternative approach to an undergraduate abstract algebra course. Finally, because of the breadth of topics it covers it would also be ideal for a graduate course for mathematics teachers.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1997

Количество страниц: 208

Добавлена в каталог: 02.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$T_{1}$-space      147
Abel      104 124
Abelian group      5 154-159
Abelianization      158
Absolute value      11
Adjoining a root      81-84
Algebraic closure      80 85
Algebraic extension      75
Algebraic integer      95
Algebraicially closed field      85
Analytic function      40 40-73
Archimedean property      9
Argument      16
Automorphism group      107
Betti number      157
Boundary      168
Boundary group      172
Brouwer degree      177
Brouwer fixed point theorem      178
Burnside problem      156
Cardano      3-4 124
Cauchy      42 63 185
Cauchy - Goursat Theorem      64-69
Cauchy - Scwarz inequality      142
Cauchy integral formula      66-69
Cauchy’s estimate      69
Cauchy’s Theorem      64-69
Caucy - Riemann equations      41 41-46
Caucy sequence property      9
Chain group      172
Characteristic of a ring      113
Classification problem      146
Closed set      145
Commutative ring      5
Commutator subgroup      158
Compact      37 148-149
Completely regular space      147
Complex analysis      36 36-73
Complex conjugate      11
Complex function      36
complex numbers      10 10-19
Complex plane      12
Complex polynomial      1 29-31
Complex variables      36 36-73
Conformal mapping      46-49
Conjugate of a polynomial      30
Conjugate of a subgroup      108
Connected      37 149
Constant polynomial      21
Constructible number      126-130
Constructions geometric      126-130
continuous      38
Contour integral      61 63
Contractible space      161
Cover      148
Covering space      166
Curve      46
Cycle group      172
Cyclic group      18 155
Degree of a polynomial      21
Degree of an extension field      8
DeMoivre’s Theorem      17-19
Differentiable      39
Differential form      55
Direct product      154
Division algebra      130
Division algorithm      24
Division ring      130
Domain      37
Double integral      52
D’Alembert      3
Elementary symmetric polynomial      89
Entire function      40
Euclidean algorithm      25
Euclidean space      140
Euler’s identity      16
Euler’s Magic Formula      16
Exact      55
Extension field      6
Extreme values theorem      32
Factor group      110-111
Factor ring      83
Fellow-traveler property      135
Field      5
Finite group      86
Free Abelian group      155
Free group      165
Fundamental group      163-167
Fundamental theorem of algebra      1
Fundamental Theorem of Arithmetic      24
Fundamental Theorem of Galois Theory      119
Galois      104
Galois extension      112-115
Galois group      115-119
Galois group of a polynomial      117
Galois theory      104-133
Gauss      3
Generators      88
Girard      3
Goursat      64
Green’s Theorem      57-59
Ground field      75
Group      86
Growth lemma      195
Harmonic function      45-46
Hausdorf space      147
Holomorphic      40
Homeomorphism      146
Homologous      172
Homology group      172-176
Homology theory      153 166-179
Homomorphism of groups      106
Homomorphism of rings      10
Homotopic      153 159-160
Homotopically equivalent      153 160
Homotopy invariant      153 161
Homotopy theory      153
Ideal      82-84
Image      110
Imaginary unit      10
Inner product      140
Inner product space      140-142
Integral domain      22
Intermediate field      75
Intermediate Value Theorem      9
Irreducible polynomial      24
Isogonal      47
Isomorphism of fields      106
Isomorphism of groups      106
Isomorphism of rings      10
Jordan curve theorem      57
Kernel      110
Lagrange      108
Lagrange’s Theorem      108
Laplace’s equation      45
Least upper bound      9
Lindemann      126
Line integral      52-61
Linear polynomial      21
Liouvilles’ Theorem      70-71
Loop      161
Magnification      48
Maximal ideal      83
Maximum principle      72
Metric space      139-140
Metrizable space      147 148
Minimal polynomial      94
Modulus      11
Morera’s theorem      71
Neighborhood      37
Netsed intervals property      9
Normal extension      112
Normal space      147
Normal subgroup      108
Normed linear space      141
Open ball      143
Open base      148
Open set      143 145
Ordered field      8
Path      161
Path-connected      163
Permutation      86-88
Piecewise continuously differentiable      185
Point-set topology      136 136-151
Polar form      14
Polynomial      21
Primitive factor      189
Primitive polynomial      94
Primitive roots of unity      18
Principal ideal      82
Proofs of      32-33 70-71 91-93 123-124 135-136 178-180 182-186 196-197 197 198 200-201
Quadratic polynomial      21
Quaternion algebra      131
Quaternions      131
Quotient group      110-111
Quotient ring      83
Real numbers      8-9
Real polynomials      29-31
Regular curve      46
Relations      88
Riemann      42
Riemann sphere      179
Ring      5
Ring of polynomials      21-22
Roots of polynomials      23 27-28
Roots of unity      18
Rouche’s Theorem      197
Ruffini      4
Second countable space      148
Separable extension      112
Separation axioms      147
Simple closed curve      56
Simple extension      77
Simple group      109
simplex      167-168
Simplicial complex      168-170
Simplicial decomposition      170
Simply connected      163
Skew field      130
Solvable by radicals      125
Solvable group      125
Splits      84
Splitting field      84-86
Star domain      188
Stereographic projection      178
Subcover      148
Subfield      6
Sylow subgroup      111
Symmetric group      86-88
Symmetric polynomial      89-91 99-102
Symmetric polynomials      89 99-102
Topological invariant      152
Topological space      145
topology      134-181 145
Torsion group      156
Totally real field      130
Transcendence of e and n      94-99
Transcendental extension      75
Transcendental number      8
Triangulation      170
Trichotomy Law      8
UFD      24
Unique factorization domain      24
UNIT      22
Universal covering space      166
Vector space      7
Wantzel      126
Winding number      134
Zero divior      22
Zero of a polynomial      23
Zero ploynomial      21
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