Chapter VII. Infinite Field Extensions

Abstract

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 $\mathrm{Gal}(L/K)$.