Index of terminology

Abelian closure : V, p. 77. Abelian extension : V, p. 77. Absolutely semi-simple endomorphism : VII, p. 42. Addition of inequalities : VI, p. 2. Adjunction (extension obtained by) : V, p. 11. Algebraic closure : V, p. 22. Algebraic element in an algebra : V, p. 16. Algebraic extension : V, p. 17. Algebraic number : V, p. 20. Algebraically closed field : V, p. 20. Algebraically dependent family, subset : V, p. 106. Algebraically disjoint extensions : V, p. 112. Algebraically free family of extensions : V, p. 115. Algebraically free family, subset : IV, p. 4, V, p. 106. Archimedean ordered group : VI, p. 35, Ex. 31. Artin’s theorem : V, p. 65. Artin-Schreier theorem : VI, p. 22. Artin-Schreier theory : V, p. 91. Associated elements : VI, p. 5.

Bezout's identity : VII, p. 2. Budan-Fourier rule : VI, p. 42, Ex. 21.

Characteristic exponent of a field : V, p. 7. Characteristic of a ring : V, p. 2. Characteristic submodule : VII, p. 67, Ex. 16. Chinese remainder theorem : VI, p. 33, Ex. 25. Closed (field, algebraically) : V, p. 20. Closed (field, separably) : V, p. 45. Closed half-line : VI, p. 28. Closed (relatively algebraically) field : V, p. 19. Closure (abelian) of a field : V, p. 77. Closure (algebraic) of a field : V, p. 22. Closure (algebraic separable) : V, p. 45. Closure (perfect) : V, p. 5. Closure (relative algebraic) : V, p. 19.

Closure (relative p-radical) : V, p. 25. Closure (relative separable algebraic) : V, p. 43. Coefficients of a formal power series : IV, p. 24. Coefficients of a polynomial : IV, p. 1. Compatible (order relation) with a group, monoid structure : VI, p. 1. Compatible (order relation) with a ring structure : VI, p. 19. Compatible (preorder relation) with a commutative monoid structure : VI, p. 3. Complementary orientation : VI, p. 29. Composite extension : V, p. 12. Composition of series : IV, p. 97, Ex. 15. Conjugacy class in $\Omega$ : V, p. 52. Conjugate elements : V, p. 52 ; Conjugate extensions : V, p. 52. Constant polynomial, term : IV, p. 1. Content : VII, p. 16. Coprime elements : VI, p. 13. Coproduct in TS(M) : IV, p. 50. Cubic extension : V, p. 10. Cyclic extension : V, p. 85. Cyclic vector space (for u) : VII, p. 29. Cyclotomic extension : V, p. 81. Cyclotomic polynomial : V, p. 81.

Decomposable module : VII, p. 23. Decomposition algebra (universal) of a polynomial : IV, p. 73. Decomposition theorem: VI, p. 11. Dedekind's theorem : V, p. 27. Degree (inseparable) of an extension : V, p. 46. Degree of an algebra over a field : V, p. 10. Degree of an algebraic element : V, p. 16. Degree (total) of a polynomial : IV, p. 3. Degree (total) of a rational fraction : IV, p. 20. Degree (separable) of an extension : V, p. 31. Derivation (partial) in a p-radical extension of height $\leq 1$ : V, p. 103. Derivative (partial) of a formal power series : IV, p. 32. Derivative (partial) of a polynomial : IV, p. 6. Descartes'rule : VI, p. 42, Ex. 21. Diagonal, diagonalizable set of endomorphisms : VII, p. 40. Diagonalizable, diagonalized algebra : V, p. 28. Direct basis : VI, p. 29. Direction vector : VI, p. 28. Direction of a half-line : VI, p. 28. Discriminant of a monic polynomial : IV, p. 81. Discriminant of a polynomial : IV, p. 83. Discriminant of a sequence of elements : V, p. 47. Disjoint (algebraically) extensions : V, p. 112.

Disjoint (linearly) extensions : V, p. 14. Divided power in TS(M) : IV, p. 45. Divided power of a series : IV, p. 96, Ex. 13. Divisibility relation : VI, p. 5. Division (euclidean) : IV, p. 10. Divisor (greatest common) : IV, p. 12, VI, p. 8. Divisor of an element : VI, p. 5. Divisors (elementary) of a module : VII, p. 24. Double root : IV, p. 15.

Eigenspace : VII, p. 30. Eigenvalue : VII, p. 30. Elementary divisor of a module : VII, p. 24. Elementary symmetric polynomial : IV, p. 61. Etale algebra : V, p. 28. Euclid's lemma : VI, p. 15. Euclidean domain : VII, p. 49, Ex. 7. Euler's identity : VI, p. 8, IV, p. 89, Ex. 6. Euler-Lagrange theorem : VI, p. 26. Expansion at the origin of a rational fraction : IV, p. 30, V, p. 39. Exponential of a formal power series : IV, p. 39. Exponential type (sequence of) : IV, p. 91, Ex. 1. Extension of a field : V, p. 9. Extension (principle of) of algebraic identities : IV, p. 18.

Finite extension : V, p. 10. Finitely generated extension : V, p. 11. Form of degree n : IV, p. 2. Formal power series : IV, p. 24. Formal power series (generalized) : IV, p. 38. Fractions (fields of rational) : IV, p. 19. Frobenius homomorphism : V, p. 4, V, p. 92.

Galois extension : V, p. 57. Galois group of an extension, of a polynomial : V, p. 58. Galois theory : V, p. 67. Gamma algebra of a module : IV, p. 92, Ex. 2. Gaussian integer : VII, p. 1. Generated (extension) : V, p. 11. Generated (Galois extension) : V, p. 57. Generated (quasi-Galois extension) : V, p. 55. Generating family of an extension : V, p. 11. Greatest common divisor (GCD) of elements : VI, p. 8. Greatest common divisor of polynomials : IV, p. 12. Greatest common divisor of principal ideals : VII, p. 3.

Half-line (closed, open) : VI, p. 28. Half-lines (opposite) : VI, p. 28. Hankel determinants : IV, p. 90, Ex. 1. Height of a p-radical element : V, p. 24. Height of a p-radical extension : V, p. 26. Height of an element of K(T) : V, p. 148, Ex. 11. Hermite interpolation formula : IV, p. 88, Ex. 13. Hermite reduced form : VII, p. 68, Ex. 21. Homogeneous component of a formal power series : IV, p. 25.

Imperfect field : V, p. 7. Imperfection (degree of) : V, p. 170, Ex. 1. Indecomposable module : VII, p. 23. Indeterminate : IV, p. 1. Index of a linear mapping : V, p. 132. Indicator (Euler's) : V, p. 79. Indivisible element : VII, p. 17. Inflation homomorphism : V, p. 70. Inseparable degree : V, p. 46. Integral ideal : VI, p. 6. Invariant factors of a linear mapping : VII, p. 22. Invariant factors of a module : VII, p. 20. Invariant factors of a submodule : VII, p. 19. Invariants (similarity) of an endomorphism : VII, p. 32. Irreducible element : VI, p. 17. Irreducible polynomial : IV, p. 13. Isobaric polynomial : IV, p. 3.

Jordan decomposition : VII, p. 43. Jordan decomposition (multiplicative) : VII, p. 46. Jordan matrix : VII, p. 35.

Kummer extension : V, p. 162, Ex. 7. Kummer theory : V, p. 88.

Lagrange interpolation formula : IV, p. 16. Law (polynomial) : IV, p. 94, Ex. 9. Leading coefficient : IV, p. 3. Least common multiple (LCM) of elements : VI, p. 8. Least common multiple of principal ideals : VII, p. 3. Lefschetz's principle : V, p. 117. Lexicographic product of ordered groups : VI, p. 7. Linear topology : IV, p. 28. Linearly disjoint extensions : V, p. 13. Linearly topologized algebra : IV, p. 28.

Logarithm of a formal power series : IV, p. 40 Logarithmic derivative : IV, p. 41. Liiroth's theorem : V, p. 149, Ex. 11.

Mac Laner's criterion : V, p. 43, V, p. 123. Maximal ordered field : VI, p. 25. Minimal polynomial of an element : V, p. 16. Minimal polynomial of an endomorphism : VII, p. 29. Minkowski's lemma : VII, p. 50, Ex. 9. Monic polynomial : IV, p. 3. Monogenous extension : V, p. 11. Monomials, IV, p. 1. Multiple factor (polynomial without) : IV, p. 14. Multiple of an element : VI, p. 5. Multiple root : IV, p. 15. Multiplicity (geometric) of an eigenvalue : VII, p. 30. Multiplicity of a root : IV, p. 15. Multiplicity of an elementary divisor : VII, p. 24.

Negative element : VI, p. 4. Negative n-vector : VI, p. 29. Negative basis : VI, p. 29. Newton polygon : V, p. 150, Ex. 2. Newton's relations : IV, p. 70, IV, p. 75. Nilpotent component : VII, p. 45. Normal basis, normal basis theorem : V, p. 73. Normal extension : V, p. 53. $Norm$ : V, p. 47.

Open half-line : VI. p. 28. Opposite half-lines : VI, p. 28. Order of a (generalized) formal power series : IV, p. 25, IV, p. 38 Order of a root : IV, p. 15. Orderable field : VI, p. 39, Ex. 8. Ordered extension : VI, p. 21. Ordered field : VI, p. 20. Ordered group, monoid : VI, p. 1. Ordered ring : VI, p. 19. Orientation of a vector space : VI, p. 29. Oriented affine space, vector space : VI, p. 29.

p-adic integers (ring of) : V, p. 96 p-basis : V, p. 98. p-basis (absolute) : V, p. 98. p-free family : V, p. 98.

p-radical closure : V, p. 25. p-radical element : V, p. 24. p-radical extension : IV, p. 19. p-torsion group : VII, p. 10. n-primary component : VII, p. 8. π-primary module : VII, p. 7. Perfect closure of a ring : V, p. 5. Perfect field : V, p. 7. Perfect ring of characteristic p : V, p. 5. Polynomial, polynomial algebra : IV, p. 1. Polynomial function : IV, p. 4. Polynomial law : IV, p. 94, Ex. 9. Polynomial mapping : IV, p. 57. Polynomial mapping (homogeneous) : IV, p. 55. Positive basis : VI, p. 29. Positive element : VI, p. 4. Positive n-vector : VI, p. 29. Power series (generalized formal) : IV, p. 24, IV, p. 38 Preordered group, monoid : VI, p. 3. Primary extension : V, p. 139. Prime field, subfield : V, p. 1. Primitive element (theorem of the) : V, p. 40. Primitive root of unity : V, p. 81. Principal fractional ideal : VI, p. 6. Principal ideal domain : VII, p. 1. Principal ideal ring : VII, p. 49, Ex. 6. Product (lexicographic) of ordered groups : VI, p. 7. Product of ordered groups : VI, p. 7. Product orientation : VI, p. 29. Puiseux's theorem : V, p. 150, Ex. 2. Pure basis : V, p. 106. Pure extension : V, p. 106. Pure submodule : VII, p. 55, Ex. 7. Pythagorean field : VI, p. 39, Ex. 8.

Quadratic extension : V, p. 10. Quadratic reciprocity (theorem of) : V, p. 166, Ex. 23. Quasi-Galois extension : V, p. 53. Quotient in euclidean division : IV, p. 11. Quotient orientation : VI, p. 29.

Rational fraction : IV, p. 19. Rational function : IV, p. 21. Reduced ring : V, p. 34. Regular algebra : V, p. 140. Regular extension : V, p. 141. Relative algebraic closure of a field in an extension : V, p. 19.

Relatively algebraically closed field in an extension field : V, p. 19. Relatively prime polynomials : IV, p. 12. Remainder in euclidean division : IV, p. 11. Representatives (system of) of irreducible elements : VII, p. 3. Restriction homomorphism : V, p. 60, V, p. 70. Resultant of two polynomials : IV, p. 76. Root of a polynomial : IV, p. 14. Root of unity : V, p. 78. Rule of signs : VI, p. 19.

Semi-simple (absolutely) component : VII, p. 45. Semi-simple (absolutely) endomorphism : VII, p. 42. Semi-simple endomorphism : VII, p. 41. Semi-simple module : VII, p. 9. Separable algebra : V, p. 119. Separable algebraic closure (relative) : V, p. 44. Separable algebraic extension : V, p. 36. Separable closure : V, p. 44. Separable degree : V, p. 31. Separable element : V, p. 39. Separable extension : V, p. 121. Separable polynomial : V, p. 37. Separably closed field : V, p. 45. Separating transcendence basis : V, p. 136. Series on E with values in F : IV, p. 96, Ex. 11. Set (\Gamma-) : V, p. 75. Sign of an element : VI, p. 20. Similar endomorphisms, matrices : VII, p. 29. Similarity invariants of on endomorphism : VII, p. 32. Simple module : VII, p. 25. Simple root : IV, p. 15. Splitting field, extension : V, p. 21. Subextension : V, p. 9. Subfield : V, p. 1. Substitutable element in a rational fraction : IV, p. 21. Substitution in a formal power series : IV, p. 29. Substitution in a polynomial : IV, p. 4. Substitution in a rational fraction : IV, p. 21. Symmetric (elementary) polynomials : IV, p. 61. Symmetric formal power series : IV, p. 67. Symmetric polynomial : IV, p. 61. Symmetric product of symmetric tensors : IV, p. 43. Symmetric rational fraction : IV, p. 67. Symmetric tensor : IV, p. 42. Symmetrized tensor : IV, p. 43.

Tangent linear mapping : IV, p. 36. Taylor's formula for formal power series : IV, p. 31.

Taylor's formula for polynomials : IV, p. 8. Term, constant term, of a formal power series : IV, p. 24. Term, constant term, of a polynomial : IV, p. 1. Trace : V, p. 47. Transcendence basis : V, p. 109. Transcendence degree : V, p. 110. Transcendental element : V, p. 16. Transcendental extension : V, p. 17. Translation homomorphism : V, p. 70. Triangularisable endomorphism : VII, p. 35. Trivial extension : V, p. 9.

Ulm-Zippin theorem : VII, p. 59, Ex. 14. Unipotent component : VII, p. 46. Unipotent endomorphism : VII, p. 46. Units of a ring : VI, p. 5. Universal decomposition of a monic polynomial : IV, p. 73.

Weight : IV, p. 3. Wilson's formula : V, p. 94.

Zero of a polynomial : IV, p. 14.