This chapter is mainly devoted to dealing with the deeper aspects of field extensions.

In Section 9.1 I prove the existence of a ‘universal extension field’ k of a field k, in which the polynomials in k[X] and even those in k[X] split into linear factors: this notion of algebraic closure generalizes the property of ℂ with respect to ℝ.

In Section 9.2, I discuss the argument which was only hinted at in Lemma 8.2.1, namely, the fact that a set of (not necessarily finite) generators of a field extension K ⊃ k can be reordered and separated so that there is an intermediate field Ktrasc such that K is an algebraic extension of Ktrasc, which is a purely transcendental extension of k. In so doing I introduce the notions of algebraic dependence and transcendental bases and show that it is possible to introduce the concept of degree for transcendental extensions, as we did for algebraic ones.

In Section 9.3 I describe the structure of finite extensions based on the above analysis and on the result that algebraic extensions of a field k are a purely inseparable extension of a separable extension of k.

In Section 9.4 I introduce another crucial concept, that of the universal field of a prime field k: this is a field which contains an isomorphic copy of any finite extension field K over k, i.e. a field in which all fields satisfying Kronecker's Model have a representation.