p-adically closed fields have normalized cross-sections, but no further study of cross-sections was needed to develop the general theory of p-adically closed fields.
When studying semialgebraic equivalence relations over Q , I grew p interested in finding a p-adically closed field without a cross-section.
Discovering none in the literature, I eventually built such a field, and Ε½ . found necessary and sufficient conditions for the existence of normalized cross-sections. These conditions appear, after the preliminary lemmas of Section 1, in Section 2, and are followed in Section 3 by a proof that any cross-section for Q extends to a cross-section for any p-adically closed p Ε½ . extension of Q . Section 4 then shows that if F, Β¨is a p-adically closed p Ε½ . Ε½ . field in which Q , Β¨does not embed, then F, Β¨has a p-adically closed p p extension without a cross-section. Section 5 refines this result by producing, among other examples, p-adically closed fields that have cross-sections but lack normalized ones. The proofs of these results will use model-theow x retic techniques explained in 3 . I am grateful to A. Macintyre and A. Pillay for helpful suggestions at an early stage of my work.
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