The Adian–Rabin theorem is a result in the field of mathematical logic, specifically in the area of decidability and the theory of algebraic structures. It addresses the properties of certain classes of roots of equations and relies on concepts from algebra and logic. In basic terms, the theorem states that for any given sequence of rational numbers, it is possible to find a computably enumerable sequence of algebraic numbers that has roots within those rational numbers.
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No Markov property of finitely presented groups is decidable by an algorithm taking an arbitrary finite group presentation as input. The proof reduces the word problem for a group to property recognition using an effective construction which collapses when an input word is trivial and embeds the input group when it is nontrivial.