This chapter establishes Gauss's Law of Quadratic Reciprocity, the theory of binary quadratic forms, and Dirichlet's Theorem on primes in arithmetic progressions.

Section 1 outlines how the three topics of the chapter occurred in natural sequence and marked a transition as the subject of number theory developed a coherence and moved toward the kind of algebraic number theory that is studied today.

Section 2 establishes quadratic reciprocity, which is a reduction formula providing a rapid method for deciding solvability of congruences $x^2\equiv m\bmod p$ for the unknown $x$ when $p$ is prime.

Sections 3–5 develop the theory of binary quadratic forms $ax^2+bxy+cy^2$, where $a,b,c$ are integers. The basic tool is that of proper equivalence of two such forms, which occurs when the two forms are related by an invertible linear substitution with integer coefficients and determinant 1. The theorems establish the finiteness of the number of proper equivalence classes for given discriminant, conditions for the representability of primes by forms of a given discriminant, canonical representatives of the finitely many proper equivalence classes of a given discriminant, a group law for proper equivalence classes of forms of the same discriminant that respects representability of integers by the classes, and a theory of genera that takes into account inequivalent forms whose values cannot be distinguished by linear congruences.

Sections 6–7 digress to leap forward historically and interpret the group law for proper equivalence classes of binary quadratic forms in terms of an equivalence relation on the nonzero ideals in the ring of integers of an associated quadratic number field.

Sections 8–10 concern Dirichlet's Theorem on primes in arithmetic progressions. Section 8 discusses Euler's product formula for $\sum_{n=1}^{\infty}n^{-s}$ and shows how Euler was able to modify it to prove that there are infinitely many primes $4k+1$ and infinitely many primes $4k+3$. Section 9 develops Dirichlet series as a tool to be used in the generalization, and Section 10 contains the proof of Dirichlet's Theorem. Section 8 uses some elementary real analysis, and Sections 9–10 use both elementary real analysis and elementary complex analysis.

This chapter studies finite-dimensional associative division algebras, as well as other finite-dimensional associative algebras and closely related rings. The chapter is in two parts that overlap slightly in Section 6. The first part gives the structure theory of the rings in question, and the second part aims at understanding limitations imposed by the structure of a division ring.

Section 1 briefly summarizes the structure theory for finite-dimensional (nonassociative) Lie algebras that was the primary historical motivation for structure theory in the associative case. All the algebras in this chapter except those explicitly called Lie algebras are understood to be associative.

Section 2 introduces left semisimple rings, defined as rings $R$ with identity such that the left $R$ module $R$ is semisimple. Wedderburn's Theorem says that such a ring is the finite product of full matrix rings over division rings. The number of factors, the size of each matrix ring, and the isomorphism class of each division ring are uniquely determined. It follows that left semisimple and right semisimple are the same. If the ring is a finite-dimensional algebra over a field $F$, then the various division rings are finite-dimensional division algebras over $F$. The factors of semisimple rings are simple, i.e., are nonzero and have no nontrivial two-sided ideals, but an example is given to show that a simple ring need not be semisimple. Every finite-dimensional simple algebra is semisimple.

Section 3 introduces chain conditions into the discussion as a useful generalization of finite dimensionality. A ring $R$ with identity is left Artinian if the left ideals of the ring satisfy the descending chain condition. Artin's Theorem for simple rings is that left Artinian is equivalent to semisimplicity, hence to the condition that the given ring be a full matrix ring over a division ring.

Sections 4–6 concern what happens when the assumption of semisimplicity is dropped but some finiteness condition is maintained. Section 4 introduces the Wedderburn–Artin radical $\mathrm{rad} R$ of a left Artinian ring $R$ as the sum of all nilpotent left ideals. The radical is a two-sided nilpotent ideal. It is 0 if and only if the ring is semisimple. More generally $R/\mathrm{rad} R$ is always semisimple if $R$ is left Artinian. Sections 5–6 state and prove Wedderburn's Main Theorem–-that a finite-dimensional algebra $R$ with identity over a field $F$ of characteristic 0 has a semisimple subalgebra $S$ such that $R$ is isomorphic as a vector space to $S\oplus\mathrm{rad} R$. The semisimple algebra $S$ is isomorphic to $R/\mathrm{rad} R$. Section 5 gives the hard part of the proof, which handles the special case that $R/\mathrm{rad} R$ is isomorphic to a product of full matrix algebras over $F$. The remainder of the proof, which appears in Section 6, follows relatively quickly from the special case in Section 5 and an investigation of circumstances under which the tensor product over $F$ of two semisimple algebras is semisimple. Such a tensor product is not always semisimple, but it is semisimple in characteristic 0.

The results about tensor products in Section 6, but with other hypotheses in place of the condition of characteristic 0, play a role in the remainder of the chapter, which is aimed at identifying certain division rings. Sections 7–8 provide general tools. Section 7 begins with further results about tensor products. Then the Skolem–Noether Theorem gives a relationship between any two homomorphisms of a simple subalgebra into a simple algebra whose center coincides with the underlying field of scalars. Section 8 proves the Double Centralizer Theorem, which says for this situation that the centralizer of the simple subalgebra in the whole algebra is simple and that the product of the dimensions of the subalgebra and the centralizer is the dimension of the whole algebra.

Sections 9–10 apply the results of Sections 6–8 to obtain two celebrated theorems—Wedderburn's Theorem about finite division rings and Frobenius's Theorem classifying the finite-dimensional associative division algebras over the reals.

This chapter continues the study of finite-dimensional associative division algebras over a field $F$, with particular attention to those that are simple and have center $F$. Section 5 is a self-contained digression on cohomology of groups that is preparation for an application in Section 6 and for a general treatment of homological algebra in Chapter IV.

Section 1 introduces the Brauer group of $F$ and the relative Brauer group of $K/F$, $K$ being any finite extension field. The Brauer group $\mathcal{B}(F)$ is the abelian group of equivalence classes of finite-dimensional central simple algebras over $F$ under a relation called Brauer equivalence. The inclusion $F\subseteq K$ induces a group homomorphism $\mathcal{B}(F)\to\mathcal{B}(K)$, and the relative Brauer group $\mathcal{B}(K/F)$ is the kernel of this homomorphism. The members of the kernel are those classes such that the tensor product with $K$ of any member of the class is isomorphic to some full matrix algebra $M_{n}(K)$; such a class always has a representative $A$ with $\dim_FA=(\dim_FK)^2$. One proves that $\mathcal{B}(F)$ is the union of all $\mathcal{B}(K/F)$ as $K$ ranges over all finite Galois extensions of $F$.

Sections 2–3 establish a group isomorphism $\mathcal{B}(K/F)\cong H^2(\mathrm{Gal}(K/F),K^{\times})$ when $K$ is a finite Galois extension of $F$. With these hypotheses on $K$ and $F$, Section 2 introduces data called a factor set for each member of $\mathcal{B}(K/F)$. The data depend on some choices, and the effect of making different choices is to multiply the factor set by a “trivial factor set.” Passage to factor sets thereby yields a function from $\mathcal{B}(K/F)$ to the cohomology group $H^2(\mathrm{Gal}(K/F),K^{\times})$. Section 3 shows how to construct a concrete central simple algebra over $F$ from a factor set, and this construction is used to show that the function from $\mathcal{B}(K/F)$ to $H^2(\mathrm{Gal}(K/F),K^{\times})$ constructed in Section 2 is one-one onto. An additional argument shows that this function in fact is a group isomorphism.

Section 4 proves under the same hypotheses that $H^1(\mathrm{Gal}(K/F),K^{\times})=0$, and a corollary makes this result concrete when the Galois group is cyclic. This result and the corollary are known as Hilbert's Theorem 90.

Section 5 is a self-contained digression on the cohomology of groups. If $G$ is a group and $\mathbb{Z} G$ is its integral group ring, a standard resolution of $\mathbb{Z}$ by free $\mathbb{Z} G$ modules is constructed in the category of all unital left $\mathbb{Z} G$ modules. This has the property that if $M$ is an abelian group on which $G$ acts by automorphisms, then the groups $H^n(G,M)$ result from applying the functor $\mathrm{Hom}_{\mathbb{Z} G}(\,\cdot\,,M)$ to the members of this resolution, dropping the term $\mathrm{Hom}_{\mathbb{Z} G}(\mathbb{Z},M)$, and taking the cohomology of the resulting complex. Section 5 goes on to show that the groups $H^n(G,M)$ arise whenever this construction is applied to any free resolution of $\mathbb{Z}$, not necessarily the standard one. This section serves as a prerequisite for Section 6 and as motivational background for Chapter IV.

Section 6 applies the result of Section 5 in the case that $G$ is finite cyclic, producing a nonstandard free resolution of $\mathbb{Z}$ in this case. From this alternative free resolution, one obtains a rather explicit formula for $H^2(G,M)$ whenever $G$ is finite cyclic. Application to the case that $G$ is the Galois group $\mathrm{Gal}(K/F)$ for a finite Galois extension gives the explicit formula $\mathcal{B}(K/F)\cong F^{\times}\big/N_{K/F}(K^{\times})$ for the relative Brauer group when the Galois group is cyclic.

This chapter develops the rudiments of the subject of homological algebra, which is an abstraction of various ideas concerning manipulations with homology and cohomology. Sections 1–7 work in the context of good categories of modules for a ring, and Section 8 extends the discussion to abelian categories.

Section 1 gives a historical overview, defines the good categories and additive functors used in most of the chapter, and gives a more detailed outline than appears in this abstract.

Section 2 introduces some notions that recur throughout the chapter—complexes, chain maps, homotopies, induced maps on homology and cohomology, exact sequences, and additive functors. Additive functors that are exact or left exact or right exact play a special role in the theory.

Section 3 contains the first main theorem, saying that a short exact sequence of chain or cochain complexes leads to a long exact sequence in homology or cohomology. This theorem sees repeated use throughout the chapter. Its proof is based on the Snake Lemma, which associates a connecting homomorphism to a certain kind of diagram of modules and maps and which establishes the exactness of a certain 6-term sequence of modules and maps. The section concludes with proofs of the crucial fact that the Snake Lemma and the first main theorem are functorial.

Section 4 introduces projectives and injectives and proves the second main theorem, which concerns extensions of partial chain and cochain maps and also construction of homotopies for them when the complexes in question satisfy appropriate hypotheses concerning exactness and the presence of projectives or injectives. The notion of a resolution is defined in this section, and the section concludes with a discussion of split exact sequences.

Section 5 introduces derived functors, which are the basic mathematical tool that takes advantage of the theory of homological algebra. Derived functors of all integer orders $\geq0$ are defined for any left exact or right exact additive functor when enough projectives or injectives are present, and they generalize homology and cohomology functors in topology, group theory, and Lie algebra theory.

Section 6 implements the two theorems of Section 3 in the situation in which a left exact or right exact additive functor is applied to an exact sequence. The result is a long exact sequence of derived functor modules. It is proved that the passage from short exact sequences to long exact sequences of derived functor modules is functorial.

Section 7 studies the derived functors of $\mathrm{Hom}$ and tensor product in each variable. These are called $\mathrm{Ext}$ and $\mathrm{Tor}$, and the theorem is that one obtains the same result by using the derived functor mechanism in the first variable as by using the derived functor mechanism in the second variable.

Section 8 discusses the generalization of the preceding sections to abelian categories, which are abstract categories satisfying some strong axioms about the structure of morphisms and the presence of kernels and cokernels. Some generalization is needed because the theory for good categories is insufficient for the theory for sheaves, which is an essential tool in the theory of several complex variables and in algebraic geometry. Two-thirds of the section concerns the foundations, which involve unfamiliar manipulations that need to be internalized. The remaining one-third introduces an artificial definition of “member” for each object and shows that familiar manipulations with members can be used to verify equality of morphisms, commutativity of square diagrams, and exactness of sequences of objects and morphisms. The consequence is that general results for categories of modules in homological algebra requiring such verifications can readily be translated into results for general abelian categories. The method with members, however, does not provide for constructions of morphisms member by member. Thus the construction of the connecting homomorphism in the Snake Lemma needs a new proof, and that is given in a concluding example.

This chapter establishes some essential foundational results in the subject of algebraic number theory beyond what was already in Basic Algebra.

Section 1 puts matters in perspective by examining what was proved in Chapter I for quadratic number fields and picking out questions that need to be addressed before one can hope to develop a comparable theory for number fields of degree greater than 2.

Sections 2–4 concern the field discriminant of a number field. Section 2 contains the definition of discriminant, as well as some formulas and examples. The main result of Section 3 is the Dedekind Discriminant Theorem. This concerns how prime ideals $(p)$ in $\mathbb{Z}$ split when extended to the ideal $(p)R$ in the ring of integers $R$ of a number field. The theorem says that ramification, i.e, the occurrence of some prime ideal factor in $R$ to a power greater than 1, occurs if and only if $p$ divides the field discriminant. The theorem is proved only in a very useful special case, the general case being deferred to Chapter VI. The useful special case is obtained as a consequence of Kummer's criterion, which relates the factorization modulo $p$ of irreducible monic polynomials in $\mathbb{Z}[X]$ to the question of the splitting of the ideal $(p)R$. Section 4 gives a number of examples of the theory for number fields of degree 3.

Section 5 establishes the Dirichlet Unit Theorem, which describes the group of units in the ring of algebraic integers in a number field. The torsion subgroup is the subgroup of roots of unity, and it is finite. The quotient of the group of units by the torsion subgroup is a free abelian group of a certain finite rank. The proof is an application of the Minkowski Lattice-Point Theorem.

Section 6 concerns class numbers of algebraic number fields. Two nonzero ideals $I$ and $J$ in the ring of algebraic integers of a number field are equivalent if there are nonzero principal ideals $(a)$ and $(b)$ with $(a)I=(b)J$. It is relatively easy to prove that the set of equivalence classes has a group structure and that the order of this group, which is called the class number, is finite. The class number is 1 if and only if the ring is a principal ideal domain. One wants to be able to compute class numbers, and this easy proof of finiteness of class numbers is not helpful toward this end. Instead, one applies the Minkowski Lattice-Point Theorem a second time, obtaining a second proof of finiteness, one that has a sharp estimate for a finite set of ideals that need to be tested for equivalence. Some examples are provided. A by-product of the sharp estimate is Minkowski's theorem that the field discriminant of any number field other than $\mathbb{Q}$ is greater than 1. In combination with the Dedekind Discriminant Theorem, this result shows that there always exist ramified primes over $\mathbb{Q}$.

This chapter develops tools for a more penetrating study of algebraic number theory than was possible in Chapter V and concludes by formulating two of the main three theorems of Chapter V in the modern setting of “adeles” and “ideles” commonly used in the subject.

Sections 1–5 introduce discrete valuations, absolute values, and completions for fields, always paying attention to implications for number fields and for certain kinds of function fields. Section 1 contains a prototype for all these notions in the construction of the field $\mathbb{Q}_p$ of $p$-adic numbers formed out of the rationals. Discrete valuations in Section 2 are a generalization of the order-of-vanishing function about a point in the theory of one complex variable. Absolute values in Section 3 are real-valued multiplicative functions that give a metric on a field, and the pair consisting of a field and an absolute value is called a valued field. Inequivalent absolute values have a certain independence property that is captured by the Weak Approximation Theorem. Completions in Section 4 are functions mapping valued fields into their metric-space completions. Section 5 concerns Hensel's Lemma, which in its simplest form allows one to lift roots of polynomials over finite prime fields $\mathbb{F}_p$ to roots of corresponding polynomials over $p$-adic fields $\mathbb{Q}_p$.

Section 6 contains the main theorem for investigating the fundamental question of how prime ideals split in extensions. Let $K$ be a finite separable extension of a field $F$, let $R$ be a Dedekind domain with field of fractions $F$, and let $T$ be the integral closure of $R$ in $K$. The question concerns the factorization of an ideal $\mathfrak{p} T$ in $T$ when $\mathfrak{p}$ is a nonzero prime ideal in $R$. If $F_{\mathfrak{p}}$ denotes the completion of $F$ with respect to $\mathfrak{p}$, the theorem explains how the tensor product $K\otimes_FF_{\mathfrak{p}}$ splits uniquely as a direct sum of completions of valued fields. The theorem in effect reduces the question of the splitting of $\mathfrak{p} T$ in $T$ to the splitting of $F_{\mathfrak{p}}$ in a complete field in which only one of the prime factors of $\mathfrak{p} T$ plays a role.

Section 7 is a brief aside mentioning additional conclusions one can draw when the extension $K/F$ is a Galois extension.

Section 8 applies the main theorem of Section 6 to an analysis of the different of $K/F$ and ultimately to the absolute discriminant of a number field. With the new sharp tools developed in the present chapter, including a Strong Approximation Theorem that is proved in Section 8, a complete proof is given for the Dedekind Discriminant Theorem; only a partial proof had been accessible in Chapter V.

Sections 9–10 specialize to the case of number fields and to function fields that are finite separable extensions of $\mathbb{F}_q(X)$, where $\mathbb{F}_q$ is a finite field. The adele ring and the idele group are introduced for each of these kinds of fields, and it is shown how the original field embeds discretely in the adeles and how the multiplicative group embeds discretely in the ideles. The main theorems are compactness theorems about the quotient of the adeles by the embedded field and about the quotient of the normalized ideles by the embedded multiplicative group. Proofs are given only for number fields. In the first case the compactness encodes the Strong Approximation Theorem of Section 8 and the Artin product formula of Section 9. In the second case the compactness encodes both the finiteness of the class number and the Dirichlet Unit Theorem.

This chapter provides algebraic background for directly addressing some simple-sounding yet fundamental questions in algebraic geometry. All the questions relate to the set of simultaneous zeros of finitely many polynomials in $n$ variables over a field.

Section 1 concerns existence of zeros. The main theorem is the Nullstellensatz, which in part says that there is always a zero if the finitely many polynomials generate a proper ideal and if the underlying field is algebraically closed.

Section 2 introduces the transcendence degree of a field extension. If $L/K$ is a field extension, a subset of $L$ is algebraically independent over $K$ if no nonzero polynomial in finitely many of the members of the subset vanishes. A transcendence basis is a maximal subset of algebraically independent elements; a transcendence basis exists, and its cardinality is independent of the particular basis in question. This cardinality is the transcendence degree of the extension. Then $L$ is algebraic over the subfield generated by a transcendence basis. Briefly any field extension can be obtained by a purely transcendental extension followed by an algebraic extension. The dimension of the set of common zeros of a prime ideal of polynomials over an algebraically closed field is defined to be the transcendence degree of the field of fractions of the quotient of the polynomial ring by the ideal.

Section 3 elaborates on the notion of separability of field extensions in characteristic $p$. Every algebraic extension $L/K$ can be obtained by a separable extension followed by an extension that is purely inseparable in the sense that every element $x$ of $L$ has a power $x^{p^e}$ for some integer $e\geq0$ with $x^{p^e}$ separable over $K$.

Section 4 introduces the Krull dimension of a commutative ring with identity. This number is one more than the maximum number of ideals occurring in a strictly increasing chain of prime ideals in the ring. For $K[X_1,\dots,X_n]$ when $K$ is a field, the Krull dimension in $n$. If $P$ is a prime ideal in $K[X_1,\dots,X_n]$, then the Krull dimension of the integral domain $R=K[X_1,\dots,X_n]/P$ matches the transcendence degree over $K$ of the field of fractions of $R$. Thus Krull dimension extends the notion of dimension that was defined in Section 2.

Section 5 concerns nonsingular and singular points of the set of common zeros of a prime ideal of polynomials in $n$ variables over an algebraically closed field. According to Zariski's Theorem, nonsingularity of a point may be defined in either of two equivalent ways—in terms of the rank of a Jacobian matrix obtained from generators of the ideal, or in terms of the dimension of the quotient of the maximal ideal at the point in question factored by the square of this ideal. The point is nonsingular if the rank of the Jacobian matrix is $n$ minus the dimension of the zero locus, or equivalently if the dimension of the quotient of the maximal ideal by its square equals the dimension of the zero locus. Nonsingular points always exist.

Section 6 extends Galois theory to certain infinite field extensions. In the algebraic case inverse limit topologies are imposed on Galois groups, and the generalization of the Fundamental Theorem of Galois Theory to an arbitrary separable normal extension $L/K$ gives a one-one correspondence between the fields $F$ with $K\subseteq F\subseteq L$ and the closed subgroups of $\mathrm{Gal}(L/K)$.

This chapter introduces aspects of the algebraic theory of systems of polynomial equations in several variables.

Section 1 gives a brief history of the subject, treating it as one of two early sources of questions to be addressed in algebraic geometry.

Section 2 introduces the resultant as a tool for eliminating one of the variables in a system of two such equations. A first form of Bezout's Theorem is an application, saying that if $f(X,Y)$ and $g(X,Y)$ are polynomials of respective degrees $m$ and $n$ whose locus of common zeros has more than $mn$ points, then $f$ and $g$ have a nontrivial common factor. This version of the theorem may be regarded as pertaining to a pair of affine plane curves.

Section 3 passes to projective plane curves, which are nonconstant homogeneous polynomials in three variables, two such being regarded as the same if they are multiples of one another. Versions of the resultant and Bezout's Theorem are valid in this context, and two projective plane curves defined over an algebraically closed field always have a common zero.

Sections 4–5 introduce intersection multiplicity for projective plane curves. Section 4 treats a line and a curve, and Section 5 treats the general case of two curves. The theory in Section 4 is completely elementary, and a version of Bezout's Theorem is proved that says that a line and a curve of degree $d$ have exactly $d$ common zeros, provided the underlying field is algebraically closed, the zeros are counted as often as their intersection multiplicities, and the line does not divide the curve. Section 5 makes more serious use of algebraic background, particularly localizations and the Nullstellensatz. It gives an indication that ostensibly simple phenomena in the subject can require sophisticated tools to analyze.

Section 6 proves a version of Bezout's Theorem appropriate for the context of Section 5: if $F$ and $G$ are two projective plane curves of respective degrees $m$ and $n$ over an algebraically closed field, then either they have a nontrivial common factor or they have exactly $mn$ common zeros when the intersection multiplicities of the zeros are taken into account.

Sections 7–10 concern Gröbner bases, which are finite generating sets of a special kind for ideals in a polynomial algebra over a field. Section 7 sets the stage, introducing monomial orders and defining Gröbner bases. Section 8 establishes a several-variable analog of the division algorithm for polynomials in one variable and derives from it a usable criterion for a finite set of generators to be a Gröbner basis. From this it is easy to give a constructive proof of the existence of Gröbner bases and to obtain as consequences solutions of the ideal-membership problem and the proper-ideal problem. Section 9 obtains a uniqueness theorem under the condition that the Gröbner basis be reduced. Adjusting a Gröbner basis to make it reduced is an easy matter. A consequence of the uniqueness result is a solution of the ideal-equality problem. Section 10 gives two theorems concerning solutions of systems of polynomial equations. The Elimination Theorem identifies in terms of Gröbner bases those members of the ideal that depend only on a certain subset of the variables. The Extension Theorem, proved under the additional assumption that the underlying field is algebraically closed, gives conditions under which a solution to the subsystem of equations that depend on all but one variable can be extended to a solution of the whole system. The latter theorem makes use of the theory of resultants.

This chapter investigates algebraic curves from the point of view of their function fields, using methods analogous to those used in studying algebraic number fields.

Section 1 gives an overview, explaining how Riemann's theory of Riemann surfaces of functions ties in with the notion of an algebraic curve and explaining how such curves can be investigated through the discrete valuations of their function fields. It is shown that what needs to be studied is arbitrary function fields in one variable over a base field. It is known that every compact Riemann surface can be viewed as an algebraic curve irreducible over $\mathbb{C}$, and thus the function fields of compact Riemann surfaces are to be viewed as informative examples of the theory in the chapter.

Section 2 introduces the notion of a divisor, which is any formal finite $\mathbb{Z}$ linear combination of the discrete valuations of the function field that are trivial on the base field, and the notion of the degree of a divisor, which is the sum of its coefficients weighted suitably. Each nonzero member $x$ of the function field gives rise to a principal divisor $(x)$, and the main result of the section is that the degree of every principal divisor is 0. This is an analog for function fields of the Artin product formula for number fields.

Section 3 contains the definition of the genus of the function field under study. The main object of study is the vector space $L(A)$ for a divisor $A$; this consists of 0 and all nonzero members $x$ of the function field such that $(x)+A$ is a divisor $\geq0$. Roughly speaking, it may be viewed as the space of functions on the zero locus of the curve whose poles are limited to finitely many points and to a certain order depending on the point. The genus is defined in terms of $\dim L(A)-\deg A$ when $A$ is a divisor that is a large multiple of the pole part of any fixed principal divisor. The main result of the section is Riemann's inequality, which says that $\dim L(A)\geq \deg A+1-g$ for all divisors $A$, where $g\geq0$ is the genus, and that $g$ is the smallest integer that works in this inequality for all divisors $A$.

Sections 4–5 concern the Riemann–Roch Theorem, which gives an interpretation of the difference of the two sides of Riemann's inequality as $\dim L(B)$ for a suitable divisor $B$ that can be defined in terms of $A$. Section 4 gives the statement and proof of the theorem, and Section 5 gives a number of simple applications.

This chapter investigates the objects and mappings of algebraic geometry from a geometric point of view, making use especially of the algebraic tools of Chapter VII and of Sections 7–10 of Chapter VIII. In Sections 1–12, $\mathbb{k}$ denotes a fixed algebraically closed field.

Sections 1–6 establish the definitions and elementary properties of varieties, maps between varieties, and dimension, all over $\mathbb{k}$. Sections 1–3 concern varieties and dimension. Affine algebraic sets, affine varieties, and the Zariski topology on affine space are introduced in Section 1, and projective algebraic sets and projective varieties are introduced in Section 3. Section 2 defines the geometric dimension of an affine algebraic set, relating the notion to Krull dimension and transcendence degree. The actual context of Section 2 is a Noetherian topological space, the Zariski topology on affine space being an example. In such a space every closed subset is the finite union of irreducible closed subsets, and the union can be written in a certain way that makes the decomposition unique. Every nonempty closed set has a meaningful geometric dimension. In affine space the irreducible closed sets are the varieties, and each variety acquires a geometric dimension. The discussion in Section 2 applies in the context of projective space as well, and thus each projective variety acquires a geometric dimension. Moreover, any nonempty open subset of a Noetherian space is Noetherian. A nonempty open subset of an affine variety is called quasi-affine, and a nonempty open subset of a projective variety is called quasiprojective. Each quasi-affine variety or quasiprojective variety has a dimension equal to that of its closure, which is a variety.

Sections 4–6 take up maps between varieties. Section 4 introduces spaces of scalar-valued functions on quasiprojective varieties—rational functions, functions regular at a point, and functions regular on an open set. The section goes on to relate these notions for the different kinds of varieties. Section 5 introduces morphisms, which are a restricted kind of function between varieties. The tools of Sections 4–5 together show that for many purposes all the different kinds of varieties can be treated as quasiprojective varieties. Section 6 introduces rational maps between varieties; these are not everywhere-defined functions, but each can be restricted to an open dense subset on which it is a morphism. Rational maps with dense image correspond to field mappings of the fields of rational functions, with the order of the mappings reversed.

Section 7 concerns singularities at points of varieties, still over the field $\mathbb{k}$. Zariski's Theorem was stated in Chapter VII for affine varieties and partly proved at that time. In the current context it has a meaning for any point of any quasiprojective variety. The section proves the full theorem, which characterizes singular points in a way that shows they remain singular under isomorphisms of varieties.

Section 8 concerns classification questions over $\mathbb{k}$ for irreducible curves, i.e., quasiprojective varieties of dimension 1. From Section 6 it is known that two irreducible curves are equivalent under rational maps if and only if their fields of rational functions are isomorphic. The main theorem of Section 8 is that each such equivalence class of irreducible curves contains an everywhere nonsingular projective curve, and this curve is unique up to isomorphism of varieties. The points of this curve are parametrized by those discrete valuations of the underlying function field that are defined over $\mathbb{k}$.

Sections 9–12 relate the general theory of Sections 1–6 to the topic of solutions of simultaneous solutions of polynomial equations, as treated at length in Chapter VIII. Section 9 treats monomial ideals in $\mathbb{k}[X_1,\dots,X_n]$, identifying their zero loci concretely and computing their dimension. The section goes on to introduce the affine Hilbert function of this ideal, which measures the proportion of polynomials of degree $\leq s$ not in the ideal. In the way that this function is defined, it is a polynomial for large $s$ called the affine Hilbert polynomial of the ideal. Its degree equals the dimension of the zero locus of the ideal. Section 10 extends this theory from monomial ideals to all ideals, again concretely computing the dimension of the zero loci, obtaining an affine Hilbert polynomial, and showing that its degree equals the dimension of the zero locus of the ideal. Section 11 adapts the theory to homogeneous ideals and projective algebraic sets by making use of the cone in affine space over the set in projective space. Section 12 applies the theory of Section 11 to address the question how the dimension of a projective algebraic set is cut down when the set is intersected with a projective hypersurface. A consequence of the theory is the result that a homogeneous system of polynomial equations over an algebraically closed field with more unknowns than equations has a nonzero solution.

Section 13 is a brief introduction to the theory of schemes, which extends the theory of varieties by replacing the underlying algebraically closed field by an arbitrary commutative ring with identity.