A discrete form of the theorem that each field endomorphism of $$ \mathbb{R}{\left( {\mathbb{Q}_{{\text{p}}} } \right)} $$ is the identity

Abstract

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.

Finitely arithmetically fixed elements of a field

“…We will start with the definition, which generalizes and gives names to the notions introduced in [7]. This definition could also be applied to rings with identity (without an identity one has to exclude the 0-homomorphism).…”

“…P r o o f. a), b) For the proof that L is a field see Theorem 1 of [7]. The rest is obvious or easy to derive from Definition 1. c) The only non obvious part is to show that L Q ⊂ L. So let r ∈ L Q ; thus there is a finite subset A ⊂ L with r ∈ A such that for any Q-arithmetical map f :…”

“…Since L is the completion of K, also G K/K is trivial. Thus Proposition 2 yields Theorems 3 and 4 of [7], but now with purely algebraic proofs.…”

“…Recently A. Tyszka [7] investigated the elements r ∈ L with the following property: there exists a finite set A ⊂ L with r ∈ A such that for any map f : A → L which behaves "like a homomorphism", one has f (r) = r. He shows that the elements with this property form a subfield of L, which is obviously fixed under all field endomorphisms of L. As main results he proves in [7] that for L ∈ {R, Q p | p ∈ P} the elements with the above property are exactly the algebraic ones. For the proof of one inclusion he uses that the theory of real closed fields (of formally p-adic fields, resp.)…”

A characterization property on field equivalent to algebraicity on Banach spaces

“…In this section, we present the general background needed to understand other sections. We begin with a presentation of the work of Tyszka in [5] which is the base of this article. Next we define the notion of automatic sequences and talk about the theorem of Christol.…”