Chapter 7 was almost entirely devoted to field extensions of finite degree and concentrated on Galois theory . However, even an introductory account should make some mention of infinite field extensions, and we shall discuss them in the present chapter, including transcendental extensions (Section 11.3) and infinite Galois theory (Section 11.8). The notion of algebraic dependence has similarities to linear dependence, which are described in abstract form in Section 11.1 and applied in Section 11.2. In addition there are concise accounts of topics that are useful in commutative ring theory and algebraic geometry, besides being of independent interest : separability (Sections 11.4 and n .s), the interactions of two or more subfields (Sections 11.6 and 11.7), applications of Galois theory (Section 11.9) and abelian extensions of finite exponent (Section 11.10).
KeywordsPrime Ideal Galois Group Dependence Relation Algebraic Closure Minimal Polynomial
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