A discrete form of the theorem that each field endomorphism of R (Q_p) is the identity

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dc.creatorTyszka, Apoloniusz
dc.date2004-01-21
dc.date2005-09-13
dc.date.accessioned2026-07-07T06:31:31Z
dc.date.available2026-07-07T06:31:31Z
dc.descriptionLet K be a field and F denote the prime field in K. Let \tilde{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) \to 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. Obviously, each field endomorphism of K is the identity on \tilde{K}. We prove: \tilde{K} is a countable subfield of K, if char(K) \neq 0 then \tilde{K}=F, \tilde{C}=Q, if each element of K is algebraic over F=Q then \tilde{K}={x \in K: x is fixed for all automorphisms of K}, \tilde{R} is equal to the field of real algebraic numbers, \tilde{Q_p}={x \in Q_p: x is algebraic over Q}.
dc.descriptionto appear in Aequationes Math., Theorem 5 provides a new characterization of \tilde{K}
dc.identifierhttps://arxiv.org/abs/math/0401285
dc.identifierhttp://arxiv.org/abs/math/0401285
dc.identifierAequationes Mathematicae 71 (2006), no. 1-2, pp. 100-108
dc.identifierdoi:10.1007/s00010-005-2801-y
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/98626
dc.subjectNumber Theory
dc.subjectCommutative Algebra
dc.subject12E99, 12L12
dc.titleA discrete form of the theorem that each field endomorphism of R (Q_p) is the identity
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