Geodesic
Undecidability of the theory of Abelian groups with an automorphism
Matematičeskie zametki, Tome 23 (1978) no. 4, pp. 515-520 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove undecidability of the elementary theories of: 1) (torsion-free) Abelian groups; 2) (Archimedean) ordered Abelian groups; 3) complete Abelian groups in the signature $\langle+,\alpha(x)=y\rangle$, where + is addition and $\alpha(x)=y$ is an automorphism of the (ordered) Abelian group. 
@article{MZM_1978_23_4_a2,
     author = {V. I. Mart'yanov},
     title = {Undecidability of the theory of {Abelian} groups with an automorphism},
     journal = {Matemati\v{c}eskie zametki},
     pages = {515--520},
     year = {1978},
     volume = {23},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1978_23_4_a2/}
}
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V. I. Mart'yanov. Undecidability of the theory of Abelian groups with an automorphism. Matematičeskie zametki, Tome 23 (1978) no. 4, pp. 515-520. http://geodesic.mathdoc.fr/item/MZM_1978_23_4_a2/