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Jerry Shurman — Geometry of the Quintic

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Название: Geometry of the Quintic

Автор: Jerry Shurman

Аннотация:

A consolidation of mature mathematical subjects like geometry, linear algebra, group theory, complex analysis and Galois theory into one source, this simple, easy-to-follow text develops deep connections between seemingly unrelated areas in mathematics. It updates Felix Klein's "Lectures on the Icosahedron and Equations of the Fifth Degree", and Peter Doyle's and Curt McMullen's "Solving the quintic by iteration." It provides an active approach to learning, and presents familiar subjects in a nonredundant, forward looking fashion.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1997

Количество страниц: 216

Добавлена в каталог: 18.10.2010

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Предметный указатель

$Aut(\widehat{C})$ (automorphisms of the Riemann sphere)      17
$C(Z)^{\Gamma}$ ( $\Gamma$ -invariant subfield of C(Z))      49
$dZ_{1}, dZ_{2}$ (differentials)      173
$F_{1,C_{n}}, F_{2,C_{n}}$ (cyclic forms)      58
$F_{1,D_{n}}, F_{2,D_{n}}, F_{3,D_{n}}$ (dihedral forms)      59
$F_{1,I}, F_{2,I}, F_{3,I}$ (icosahedral forms)      61
$F_{1,O}, F_{2,O}, F_{3,O}$ (octahedral forms)      60 61
$F_{1,T}, F_{2,T}, F_{3,T}$ (tetrahedral forms)      60
$f_{11}$ (model for the Brioschi iteration)      179
$f_{C_{n}}$ (cyclic invariant)      66
$f_{D_{n}}$ (dihedral invariant)      66
$f_{I}$ (icosahedral invariant)      66
$f_{O}$ (octahedral invariant)      66
$f_{T}$ (tetrahedral invariant)      66
$f_{\Gamma}$ (normalized generator of $C(Z)^{\Gamma}$ )      49
$GL_{2}(C)$ (general linear group)      17
$O_{3}(R)$ (orthogonal group)      21
$PGL_{2}(C)$ (projective general linear group)      18
$PSL_{2}(C)$ (projective special linear group)      18
$PSU_{2}(C)$ (projective special unitary group)      22
$PU_{2}(C)$ (projective unitary group)      22
$P^{1}(C)$ (complex projective line)      13
$P^{n}(C)$ (complex projective space)      120
$Rot(\widehat{C})$ (rotations of the Riemann sphere)      21
$SL_{2}(C)$ (special linear group)      18
$SO_{3}(R)$ (special orthogonal group)      21
$SU_{2}(C)$ (special unitary group)      22
$S^{2}$ (sphere)      1
$s_{I}, t_{I}$ (icosahedral generators)      44
$s_{j}$ (power sum)      107
$s_{n}$ (cyclic and dihedral generator)      43
$s_{O}, t_{O}$ (octahedral generators)      44
$s_{T}, t_{T}$ (tetrahedral generators)      43
$t_{D}$ (second dihedral generator)      43
$U_{2}(C)$ (unitary group)      22
$\bar{ }$ (algebraic closure of a field)      73
$\chi_{1,C_{n}}, \chi_{2,C_{n}}, \chi_{C_{n}}$ (cyclic characters)      59
$\chi_{1,D_{n}}, \chi_{2,D_{n}}, \chi_{3,D_{n}}, \chi_{D_{n}}$ (dihedral characters)      59
$\chi_{1,O}, \chi_{2,O}, \chi_{3,O}, \chi_{O}$ (octahedral characters)      61
$\chi_{1,T}, \chi_{2,T}, \chi_{3,T}, \chi_{T}$ (tetrahedral characters)      60
$\Delta$ (discriminant)      104
$\Gamma_{I}$ (icosahedral group)      44
$\Gamma_{O}$ (octahedral group)      44
$\Gamma_{T}$ (tetrahedral group)      43
$\mathcal{O}_{1}$ (vertex orbit)      41
$\mathcal{O}_{2}$ (face-center orbit)      41
$\mathcal{O}_{3}$ (mid-edge orbit)      41
$\phi_{Z}$ (conjugating transformation for the Brioschi iteration)      181
$\sigma_{j}$ (elementary symmetric function)      104
$\widehat{C}$ (Riemann sphere)      8
$\widehat{W}$ (renormalized icosahedral invariant)      185
$\widetilde{F}_{1,O}, \widetilde{F}_{2,O}$ (rotated octahedral forms)      94 96
$\widetilde{r}$ (tetrahedral invariant)      98
$\widetilde{s}$ (Brioschi root)      96
$\widetilde{\Gamma}_{T}$ (rotated tetrahedral group)      93
$\zeta_{m}$ ( $m^{th}$ root of unity)      43
$^{rad}$ (radical closure of a field)      139
$^{t}$ (transpose of a matrix)      20
$_{*}$ (dehomogenization of a form)      15
* (adjoint of a matrix)      21
* (homogenization of a polynomial)      15
* (pullback of a differential form)      174
* (pullback of a fractional linear transformation)      79
/ (field extension)      71
0-form      173
1-form      173
2-form      173
Adjoint of a matrix      21
Affine space      120
Algebraic closure      73
Algebraic element over a field      71
Algebraic extension      71
Algebraic geometry      155
Algebraic mapping      14
Algebraic set      120
Algebraic set, associated to an ideal      155
Algebraically independent elements      103
Anti-homomorphism      80
Att( ) (attractor of a rational map)      163
Attractor of a rational function      163
Aut ( ) (automorphism group of an extension)      75
Automorphism of a field      75
Automorphism of the Riemann sphere      17
Auxiliary irrationality      115
Axiom of Choice      136
Ball in $R^{n}$       2
Birational equivalence of varieties      157
Bounded set in $R^{n}$       3
Brioschi parameter      96
Brioschi quintic      96
Brioschi root      96
C (canonical surface)      123
Canonical surface      123
CHARACTER      51
Characteristic p      70
Characteristic zero      70
Closed set in $R^{n}$       3
Coefficient field      104
Collineation      121
Commutator      62
Commutator subgroup of a group      62
Compact topological space      3
Complex projective line      13
Complex projective space      120
Congruence class of a matrix      124
Congruent points under a group action      30
Congruent sets under a group action      31
Conjugate elements of a group      19
Conjugate subgroups of a group      26
Conjugating conformal map      149 165
Connected topological space      160
Constructible extension by iteration      169
Constructible extension by radicals      82
Continuity in Euclidean space      2
Continuity,topological definition of      3
Continuity-Compactness Theorem      4
Contravariant map      80
Coordinate ring of an algebraic set      155
Covariant      56
Covering group      160
Critical point of a rational function      182
Critically finite rational function      183
cube      33
Cycle of a rational function      182
Cyclic characters      59
Cyclic extension      83
Cyclic forms      58
Cyclic invariant      66
d (form differentiation operator)      174
Decision-free iteration      148
Degenerate orbit-form      55
Degree of a hypersurface      120
Degree of an extension      71
Dehomogenization operator      15
Dense subset of a topological space      156
Depressed polynomial      113
Differential      173
Differential form      173
Digon      45
Dihedral characters      59
Dihedral forms      59
Dihedral invariant      66
Dihedral syzygy      59
Dimension Lemma      135
Discrete topological space      160
Discriminant      104
dodecahedron      37
Dominant rational map of varieties      157
Dual 1-form of a rational function      176
Dual of a Platonic solid      32

Eisenstein's criterion      74
Elementary symmetric functions      104
Elliptic curve      154
Elliptic element of $PSL_{2}(C)$       29
Embedding lemma      144
Euclidean algorithm      70
Euclidean ring      70
Exceptional form      63
Faithful group action      30
Field extension      71
Finite extension      71
First Isomorphism Theorem, group theory      19
First Isomorphism Theorem, ring theory      70
Flag      32
FORM      14
Form differentiation operator      174
Form resolvent      94
Form-class      52
Fractional linear transformation      17
Full orbit-form      54
Function field class      159
Function field of a variety      156
Fundamental theorem of symmetric functions      106
g (golden ratio)      34
Gal( / ) (Galois group of an extension)      76
Galois closure      83
Galois correspondence      76
Galois extension      76
Galois group      76
Gauss' lemma      74 81
General convergence of Newton's method for $n^{th}$ roots      148
General linear group      17
Generally convergent algorithm      163
Golden ratio      34
Group action      30
Heine — Borel theorem      3
Hermitian matrix      27
Hessian covariant      56
High school error      71
Homeomorphism      3
Homogeneous coordinates      120
Homogeneous ideal      158
Homogeneous polynomial      14
Homogenization operator      15
Hyperplane      120
Hypersurface      120
I( ) (ideal associated to a set)      155
Icosahedral forms      61
Icosahedral invariant      66
Icosahedral syzygy      61
icosahedron      34
Ideal associated to a set      155
Induced topology      3
Inner product on $C^{2}$       27
Inner product properties      27
Invariant form      51
Invariant function      50
Inverse image      2
Iteration Criterion (over C)      170
Iteration tower over a field      169
Iterative algorithm      162
Jacobian covariant      56
K( ) (function field of a variety)      156
Kernel of a group action      30
Klein four-group      44
Kronecker's theorem      139
K[ ] (coordinate ring of an algebraic set)      155
Lagrange Lemma      84
Length of an orbit      30
Line in projective space      121
Linear Independence of Automorphisms      83
Local coordinate function      9
Lueroth's theorem      142
Map over a field      75
Meromorphic function      9
Minimal polynomial      71
Model for an iteration      149 165
Moebius group      170
Multiset      52
Natural irrationality      115
Nearly solvable group      170
Neighborhood      7
Newton's identities      108
Newton's method      147
Newton's method for $n^{th}$ roots of unity      148
Normal extension      75
Normal function with respect to a group      176
Normal function with respect to an automorphism      176
Octahedral characters      61
Octahedral forms      60 61
Octahedral invariant      66
Octahedral syzygy      61
octahedron      33
Open set in $R^{n}$       2
Open set in a topological space      3
Open set properties of $R^{n}$       2
Open set properties of a topological space      3
Orbit      30
Orbit space      66
Orbit-stabilizer theorem      30
Order of a meromorphic function      10
Orthogonal group      21
Orthogonal matrix      21
Out( ) (output of an iteration)      163
Output field of an algorithm      164
Output of an algorithm      163
Platonic solid      32
Polynomial on an algebraic set      155
Positive matrix      27
Power sums      107
Prime subfield of a field      70
Primitive polynomial      81
Principal polynomial      114
Projective general linear group      18
Projective special linear group      18
Projective special unitary group      22
Projective transformation      121
Projective unitary group      22
Pullback      79
Pullback of a differential form      174
Pullback of a rational map      157
Purely iterative algorithm      162
Quadric hypersurface      121
Quasi-projective variety      158
Quotient topology      4
R( , ) (resultant of two polynomials)      110
Radical closure of a field      139
Radical Criterion (over C)      83
Radical element over a field      139
Radical reduction of a polynomial      139
Rational function on a variety      156
Rational map of varieties      157
Repelling fixed point of a rational function      182
Resolvent      91
Resultant      110
Riemann sphere      8
Rigid family of rational maps      149 165
Rigid motion of $R^{3}$       25
Root field      72
Root tower over a field      82
Rotated tetrahedral group      93
Rotation of the Riemann sphere      21
Rotation of the sphere      20
Running      4 12 187
Second isomorphism theorem      19
Self-adjoint matrix      27
Separable element over a field      75
Separable extension      75
Separable polynomial      75
Simple group      89

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