[This article](https://link.springer.com/article/10.1007/BF03022866) states that in 1930 Skolem independently (of Presburger) published quantifier-eliminations for linear algebra over the integers.


I have checked the Ω-Bibliography of Mathematical Logic and was not able to find such a result.


According to Hao Wang's account in Skolem's selected works in logic this is also not the topic of Skolem's "Über einige Satzfunktionen in der Arithmetik" which is quoted in the article. The paper is also from 1931 not 1930.


Does anybody know about this paper of Skolem?



Skolem's version of quantifier elimination for linear algebra over the integers
I don’t have access to Skolem’s treatise, but according to its reviews by [Gödel](https://zbmath.org/?q=an:0002.00302) and [Ackermann](https://zbmath.org/?q=an:57.1320.03), he proves quantifier elimination for a somewhat unusual system with variables ranging over integers, function symbols for addition, multiplication by rational constants, and $\lfloor x\rfloor$, and the ordering predicate (so, apparently, values of compound terms may be rationals, unlike variables). This is equivalent to Presburger arithmetic.


Concerning the discrepancy in years, the paper is identified as “Skr. Norske Vid.-Akad., Oslo, Math.-Naturv. Kl. 1930, No. 7, 1–28 (1931)”. I interpret this to mean that the paper was published in 1931, but nominally in a 1930 volume.


