and F(x)
0.2 Arithmetic of cardinal numbers
0.3 Ordinal and cardinal numbers
1.1 Definition and examples of categories
1.2 Some basic categorical concepts
1.3 Functors and natural transformations
1.6 The horn functors. Representable functors
2.3 Homomorphisms and congruences
2.4 The lattice of congruences. Subdirect products
2.10 Internal characterization of varieties
3.1 The categories R-mod and mod-R
3.2 Artinian and Noetherian modules
3.3 Schreier refinement theorem. Jordan-Hölder theorem
3.5 Completely reducible modules
3.6 Abstract dependence relations. Invariance of dimensionality
3.7 Tensor products of modules
3.11 Injective modules. Injective hull
3.13 The Wedderburn-Artin theorem for simple rings
3.14 Generators and progenerators
3.15 Equivalence of categories of modules
4 BASIC STRUCTURE THEORY OF RINGS
4.1 Primitivity and semi-primitivity
4.5 Structure theory of algebras
4.6 Finite dimensional central simple algebras
5 CLASSICAL REPRESENTATION THEORY OF FINITE GROUPS
5.1 Representations and matrix representations of groups
5.3 Application of the representation theory of algebras
5.4 Irreducible representations of Sn
5.5 Characters. Orthogonality relations
5.6 Direct products of groups. Characters of abelian groups
5.7 Some arithmetical considerations
5.10 Properties of induction. Frobenius reciprocity theorem
5.11 Further results on induced modules
5.12 Brauer’s theorem on induced characters
5.13 Brauer’s theorem on splitting fields
6 ELEMENTS OF HOMOLOGICAL ALGEBRA WITH APPLICATIONS
6.1 Additive and abelian categories
6.3 Long exact homology sequence
6.13 Koszul’s complex and Hilbert’s syzygy theorem
7 COMMUTATIVE IDEAL THEORY: GENERAL THEORY AND NOETHERIAN RINGS
7.4 Localization at the complement of a prime ideal. Local-global relations
7.5 Prime spectrum of a commutative ring
7.8 Rank of projective modules
7.11 Commutative artinian rings
7.12 Affine algebraic varieties. The Hilbert Nullstellensatz
7.14 Artin-Rees lemma. Krull intersection theorem
7.15 Hilbert’s polynomial for a graded module
7.16 The characteristic polynomial of a noetherian local ring
7.18 I-adic topologies and completions
8.1 Algebraic closure of a field
8.2 The Jacobson-Bourbaki correspondence
8.4 Crossed products and the Brauer group
8.7 Separability and normality
8.8 Separable splitting fields
8.13 Transcendency bases for domains. Affine algebras
8.15 Separability for arbitrary extension fields
8.17 Galois theory for purely inseparable extensions of exponent one
8.18 Tensor products of fields
8.19 Free composites of fields
9.3 Absolute values on and F(x)
9.5 Finite dimensional extensions of complete fields. The archimedean case
9.7 Valuation rings and places
9.8 Extension of homomorphisms and valuations
9.9 Determination of the absolute values of a finite dimensional extension field
9.10 Ramification index and residue degree. Discrete valuations
9.13 Totally disconnected locally compact division rings
9.14 The Brauer group of a local field
9.15 Quadratic forms over local fields
10.1 Fractional ideals. Dedekind domains
10.2 Characterizations of Dedekind domains
10.3 Integral extensions of Dedekind domains
10.4 Connections with valuation theory
10.5 Ramified primes and the discriminant
10.6 Finitely generated modules over a Dedekind domain
11.3 Totally positive elements
11.4 Hilbert’s seventeenth problem
11.5 Pfister theory of quadratic forms
11.6 Sums of squares in R(x1,…,xn), R a real closed field