Defining Transcendentals in Function Fields.

Journal article

Given any field K, there is a function field F/K in one variable containing definable transcendentals over K, i,e., elements in F/K first-order definable in the language of fields with parameters from K. Hence, the model-theoretic and the field-theoretic relative algebraic closure of K in F do not coincide. E.g., if K is finite, the model-theoretic algebraic closure of K in the rational function field K(t) is K(t). For the proof, diophantine ∅-definability of K in F is established for any function field F/K in one variable, provided K is large, or K x /(K x)n is finite for some integer n > 1 coprime to char K.

Authors: Koenigsmann, J

• Publication status: Published
• Journal: J. Symb. Log.
• Volume: 67
• Issue: 3
• Pages: 947-956
• Publication date: 2002-01-01
• DOI: 10.2178/jsl/1190150142
• EISSN: 1943-5886
• ISSN: 0022-4812
• Language: English