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Hodge W.V.D., Pedoe D. — Methods of Algebraic Geometry: Volume 1
Hodge W.V.D., Pedoe D. — Methods of Algebraic Geometry: Volume 1

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Название: Methods of Algebraic Geometry: Volume 1

Авторы: Hodge W.V.D., Pedoe D.


This classic work (first published in 1947), in three volumes, provides a lucid and rigorous account of the foundations of modern algebraic geometry. The authors have confined themselves to fundamental concepts and geometrical methods, and do not give detailed developments of geometrical properties but geometrical meaning has been emphasized throughout. This first volume is divided into two parts. The first is devoted to pure algebra: the basic notions, the theory of matrices over a non-commutative ground field and a study of algebraic equations. The second part is in n dimensions. It concludes with a purely algebraic account of collineations and correlations.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1953

Количество страниц: 440

Добавлена в каталог: 18.11.2009

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Предметный указатель
$PN^{l}_{n}(K)$ or left-hand projective number space of dimension n over K      176
$PN^{r}_{n}(K)$ or right-hand projective number space of dimension n over K      176
$P^{l}_{n}(K)$ or left-hand projective space of dimension n over K      185
$P^{r}_{n}(K)$ or right-hand projective space of dimension n over K      178
$\lambda$-matrices, canonical forms for      88 92
$\lambda$-matrices, elementary transformations on      88
$\lambda$-matrices, equivalence of      88
$\lambda$-matrix      86
$\lambda$-matrix, elementary divisors of      92
$\lambda$-matrix, invariant factors of      92
(n+1)-tuple      176
(n+1)-tuples, equivalent      176
Abelian or commutative group      3
Abelian or commutative group of operations on a matrix      73
Addition of points on a line      253
Additive group      3
Adjugate matrix      81
Adjunction of algebraic element      103
Adjunction of indeterminate      19
Algebra of points on a line      253
Algebraic closure of a field      96 112
Algebraic dependence      108
Algebraic element of extension field      102
Algebraic extension over a field      106
Algebraic extension, primitive element of      126
Algebraic extension, simple      101
Algebraic function field      107
Algebraic function, independence      108
Algebraic function, point of projective space      179
Algebraic functions      107
Algebraic functions, differentiation of      128
Algebraically closed field      96 112
Allowable coordinate systems      178
Allowable transformations      59
Allowable transformations of restricted type      367
Alternating group      5
Apolar vectors      69
Associated collineation      365
Associated coordinate systems      186
Associative law for multiplication of matrices      52
Associative law of composition      2
Basis for extensions of finite degree      105
Basis for ideal      142
Basis for linear set      45
Basis for linear space      183
Basis for linear space in incidence space      209
Basis for linear space, derived from Grassmann coordinates      300
Basis theorem for Grassmann coordinates      315
Basis Theorem, Hilbert's      142
Basis, finite algebraic      108
Basis, minimal algebraic      108
Bilinear correspondence      365
Bilinear form      365
Binary form      148
Binary forms, resultant of two      150
Block matrices      53
Canonical form for $\lambda$-matrices      88 92
Canonical form for collineations      331
Canonical form for correlations      411
Canonical form for matrices      60 62
Canonical form for null-polarity      380
Canonical form for polarity      372
Canonical form for projective transformation      323
Canonical form for skew-symmetric bilinear form      378 380
Canonical form for symmetric bilinear form      368 372
Chain of perspectivities or projectivity      217
Characteristic equation of matrix      95
Characteristic function of matrix      95
Characteristic matrix      95
Characteristic matrix, polynomial of algebraic element      102
Characteristic of ring      14
Coefficients of polynomial      20
Cofactor of element in determinant      77
Collinear points      191
Collineations      327
Collineations as product of null-polarities      383
Collineations as product of null-polarity and polarity      389
Collineations as product of polarities      376
Collineations with only one united point      344
Collineations, associated with correlation      365
Collineations, canonical form      331 332
Collineations, cyclic      352
Collineations, fundamental space corresponding to latent root      336
Collineations, in [3]      350
Collineations, induced in linear space      342 343
Collineations, non-special      336
Collineations, particular, with only one fundamental space      356
Collineations, Segre symbol for      350
Collineations, similar      328
Collineations, singular      358
Collineations, special case      336
Collineations, united points of      334
Column matrix printed as row matrix      50
Column of matrix as right-hand vector      66
Column of matrix, rank of matrix      66
Commutative or Abelian group      3
Complement of matrix      72
Complete factorisation of element in integral domain      29
Completely reducible polynomial      112
Compound matrix      291
Concurrent lines      191
Condition for commutative field      202
Conjugate points in polarity      367
Conjugate spaces in polarity      375
Consequences of assuming Pappus' Theorem      274
Construction for $O + U(\alpha + \beta)$      197
Construction for $O + U\alpha\beta$      199
Coordinate of point in a scale      259
Coordinate systems, allowable      178
Coordinate systems, associated      186
Coordinate systems, related      283
Coordinate systems, scale      283
Coordinates of a projected space      308
Coordinates of point in a reference system      262
Coordinates, Grassmann, of linear space      289
Coordinates, set of, for point in projective space      178
Correlation between distinct spaces      362
Correlation in [n]      364
Correlation, canonical forms for general correlation      418
Correlation, canonical forms for null-polarity      380
Correlation, canonical forms for polarity      372
Correlation, canonical forms for simple correlation      394
Correlation, exceptional points      362
Correlation, geometrical loci connected with      421
Correlation, non-singular      426
Correlation, permissible transformations for reduction of      399
Correlation, simple, geometrical properties of      419
Correlation, singular      422
Correlation, singular of first and second kind      423
Cyclic collineations      352
Decomposition of element into factors      28
Degree of a polynomial      20
Degree of extension of finite degree      105
Delta symbols used in determinant theory      74
Dependence, algebraic      108
Dependence, linear      43
Derivative of a polynomial      119
Derivative of a rational function      120
Desargues' theorem      191
Desargues' Theorem in an incidence space      213
Desargues' Theorem, assuming Pappus' Theorem      272
Desargues' Theorem, converse, assuming the theorem      196
Desargues' Theorem, general case      194
Determinant of matrix product      78
Determinant of skew matrix      80
Determinant, cofactor of element in      77
Determinant, effect of interchange of rows      80
Determinant, Laplace's expansion of      75
Determinants      73
Determinants, value of certain      79
Diagonal points of a quadrangle      244
Diagonals of a quadrangle      244
Differentiation of algebraic functions      128
Differentiation of polynomials      119
Dimension of a linear set      45
Dimension of extension over a field      110
Displacement on a line      356
Distributive law for elements in a ring      6
Distributive law for multiplication of matrices      52
Division algorithm      24
Divisor of a polynomial      25
Divisor of zero, right-hand, left-hand      13
Dual correlation      365
Dual correlation, Grassmann coordinates      292
Dual correlation, projective transformation      323
Dual correlation, relationship between solutions of linear equations      70
Dual correlation, spaces      188
Dual correlation, theorems      190
Duality, principle of      190
Elementary divisors of $\lambda$-matrix      92
Elementary transformations      63
Elementary transformations on $\lambda$-matrices      88
Envelope incidence-quadric      427
Epsilon symbols used in determinant theory      73
Equation of projectivity on a line      281
Equations in several unknowns      159
Equations in several unknowns of a linear space      188
Equations in several unknowns, non-homogeneous linear      85
Equations, homogeneous linear      68
Equations, solution by determinants      84
Equivalence relation      10
Equivalent $\lambda$-matrices      88
Equivalent (n+1)-tuples      176
Equivalent extensions of a commutative field      101
Equivalent factorisations      29
Equivalent primes in a u.f.d.      30
Equivalent rings      10
Equivalent solutions of homogeneous equations      140
Even permutation      5
Exceptional points of a correlation      362
Exceptional points of a singular projective transformation      322
Exchange theorem for linear sets      46
Exchange theorem in projective space      182
Exchange theorem, generalised form      108
Extensions of a commutative field      101
Extensions of finite degree      105
Extensions of projective number space      178
Extensions of projective space      179
Extensions, equivalent, of a commutative field      101
Factor of element in integral domain      28
Factorisation in an integral domain      27
Factorisation in polynomial rings      33
Factorisation of polynomials      111
Factorisation, complete      29
Factorisations, equivalent      29
Field      13
Field obtained by adjunction of algebraic element      103
Field obtained by adjunction of transcendental element      104
Field of characteristic p      37
Field of coefficients of set of equations      146
Field of geometry algebraically closed      274
Field of geometry commutative      272
Field of geometry in an incidence space      259
Field of geometry without characteristic      273
Field of rational functions of $x_{1},...,x_{r}$      39
Field of rational functions of x      38
Field, algebraic closure      96 112
Field, algebraically closed      96 112
Field, commutative      13 272
Field, example of non-commutative      39
Field, formed by points on a line      258
Field, ground      41
Field, root, of a polynomial      114
Fields, examples of      37
Finite algebraic basis      108
Finite geometries      205
FORM      140
Form, binary      148
Forms, canonical      see "Canonical form"
Fundamental projectivities on a line      278
Fundamental space corresponding to infinite latent root of correlation      420
Fundamental space corresponding to latent root of collineation      336
Fundamental space corresponding to zero latent root of correlation      419
Fundamental space, rule for its determination      336
Fundamental spaces, their join      336
Fundamental theorem of algebra      112
General correlation      395
Generic linear space      315
Geometrical loci connected with correlations      421
Geometrical properties of a simple correlation      419
Grassmann coordinates      288
Grassmann coordinates of linear space      289
Grassmann coordinates of projected space      308
Grassmann coordinates, basis theorem      315
Grassmann coordinates, determine basis for linear space      300
Grassmann coordinates, dual      293
Grassmann coordinates, elementary properties      297
Grassmann coordinates, intersections and joins      304
Grassmann coordinates, linear space with given      312
Grassmann coordinates, quadratic p-relations      309
Grassmann coordinates, relation between, and dual      294
Grassmann coordinates, three-term p-relations      311
Ground field      41
Group      2
Group of operations on a matrix      73
Group of projective transformations      177
Group, additive      3
Group, alternating      5
Group, commutative or Abelian      3
Group, inverse of element in      4
Group, non-commutative      3
Group, subgroup      5
Group, symmetric      5
Group, unity of      4
Group, zero of      4
Harmonic conjugates      224
Harmonic conjugates, special case      229
Harmonically conjugate pairs      230
Highest common divisor of two polynomials      26
Highest factor of elements of integral domain      31
Hilbert's basis theorem      142
Hilbert's zero-theorem      165
Homogeneous equations      140
Homogeneous linear equations      68
Homographic ranges      217
Homology      356
Homology, involutory      356
Ideal      11
Ideal, basis for      142
Ideal, defines equivalence relation in rings      12
Ideal, improper      12
Ideal, prime      168
Ideal, resultant      166
Ideal, right-hand, left-hand      11
Ideal, unit      12
Ideals associated with sets of equations      140
Improper ideal      12
Incidence quadric of correlation      427
Incidence space as a $P^{l}_{n}(K)$      272
Incidence space as a $P^{r}_{n}(K)$      259
Incidence space of n dimensions      211
Incidence, Propositions of      186 208
Independence, algebraic      108
Independence, linear      43
Independent indeterminates over a field      105
Indeterminate coefficients, system of equations with      146
Indeterminate, adjunction of      19
Indeterminates, independent over a field      106
Infinity, properties of      280
Integral domain      13
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