Model completeness for finitely ramified Henselian fields valued in a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℤ</mml:mi> </mml:math> -group and for pseudo-algebraically closed fields

We prove that the theory of a finitely ramified Henselian valued field of characteristic zero with perfect residue field of positive characteristic whose value group is a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℤ</mml:mi> </mml:math> -group is model-complete in the language of rings if the theory of its residue field is model-complete in the language of rings. This extends results of Ax–Kochen [4], Macintyre [15], Ziegler [22], Basarab [5], and Prestel–Roquette [17]. We also prove that the theory of a perfect pseudo-algebraically closed (PAC) field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> such that the absolute Galois group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">Gal</mml:mi> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is pro-cyclic is model-complete in the language of rings if and only if every finite algebraic extension of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> is generated by elements that are algebraic over the prime subfield of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> . From these we deduce that every infinite algebraic extension of the field of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -adic numbers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℚ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> with finite ramification is model-complete in the language of rings. Our proofs of model completeness for Henselian fields use only basic model-theoretic and algebraic tools including Cohen’s structure theorems for complete local rings and basic results on coarsenings of valuations. These enable us to obtain short proofs of model completeness in the language of rings without any need to extend the ring language. Our proofs on PAC fields use the model theory of the absoute Galois group dual to the field and elementary invariants given by Cherlin–van den Dries–Macintyre [8] for the theory of PAC fields generalizing Ax’s results for the pseudo-finite case [3].