We define a computable function f from positive integers to positive integers. We formulate a hypothesis which states that if a system S of equations of the forms xi· xj = xk and xi + 1 = xi has only finitely many solutions in non-negative integers x1, . . . , xi, then the solutions of S are bounded from above by f (2n). We prove the following: (1) the hypothesis implies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (3) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (4) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, thenMis computable.

Let F ⊆ R denote the field of numbers which are constructible by means of ruler and compass. We prove that: (1) if x, y ∈ R n (n > 1) and |x − y| is an algebraic number then there exists a finite set S xy ⊆ R n containing x and y such that each map from S xy to R n preserving all unit distances preserves the distance between x and y; if x, y ∈ F n then we can choose S xy ⊆ F n , (2) only algebraic distances |x − y| have the property from item (1), (3) if X 1 , X 2 , ..., X m ∈ R n (n > 1) lie on some affine

A discrete form of the theorem that each field endomorphism of R (Q p ) is the identity Apoloniusz TyszkaSummary. Let K be a field and F denote the prime field in K . Let K denote the set of all r ∈ K for which there exists a finite set, satisfies also f (r) = r. Obviously, each field endomorphism of K is the identity on K . We prove: K is a countable subfield of K ; if char(K ) = 0, then K = F ; C = Q, R is equal to the field of real algebraic numbers, Qp is equal to the field {x ∈ Qp : x is algebraic over Q}. Subject Classification (2000). Primary: 12E99, 12L12. MathematicsKeywords. Field endomorphism, field endomorphism of Qp, field endomorphism of R, p-adic number that is algebraic over Q, real algebraic number.

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