On \(\emptyset\)-definable elements in a field
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Publication:2642842
zbMATH Open1126.03040arXivmath/0502565MaRDI QIDQ2642842
Publication date: 5 September 2007
Published in: Collectanea Mathematica (Search for Journal in Brave)
Abstract: Let K be a field and ilde{K} denote the set of all r in K for which there exists a finite set A(r) with {r} subseteq A(r) subseteq K such that each mapping f:A(r) o K that satisfies: if 1 in A(r) then f(1)=1, if a,b in A(r) and a+b in A(r) then f(a+b)=f(a)+f(b), if a,b in A(r) and a cdot b in A(r) then f(a cdot b)=f(a) cdot f(b), satisfies also f(r)=r. We prove: ilde{K} is a subfield of K, ilde{K}={x in K: {x} is existentially first-order definable in the language of rings without parameters}, if some subfield of K is algebraically closed then ilde{K} is the prime field in K, some elements of ilde{K} are transcendental over Q (over R, over Q_p) for a large class of fields K that are finitely generated over Q (that extend R, that extend Q_p), if K is a Pythagorean subfield of R, t is transcendental over K, and r in K is recursively approximable, then {r} is emptyset-definable in (K(t),+,cdot,0,1), if a real number r is recursively approximable then {r} is existentially emptyset-definable in (R,+,cdot,0,1,U) for some unary predicate U which is implicitly emptyset-definable in (R,+,cdot,0,1).
Full work available at URL: https://arxiv.org/abs/math/0502565
finitely generated field extensionMordell-Faltings theoremexistentially definable elementrecursively approximable real numbertranscendental element
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