Additive Polynomials and Their Role in the Model Theory of Valued Fields

Details Additive Polynomials and Their Role in the Model Theory of Valued Fields Additive Polynomials and Their Role in the Model Theory of Valued Fields

2010-03-29

arxiv.org/abs/1003.5683v1

Lecture Notes in Logic 26 (2006), 160-203

Mathematics

Commutative Algebra

44 pages; Proceedings of the Workshop and Conference on Logic, Algebra, and Arithmetic, held October 18-22, 2003.

Scientific paper

We discuss the role of additive polynomials and $p$-polynomials in the theory of valued fields of positive characteristic and in their model theory. We outline the basic properties of rings of additive polynomials and discuss properties of valued fields of positive characteristic as modules over such rings. We prove the existence of Frobenius-closed bases of algebraic function fields $F|K$ in one variable and deduce that $F/K$ is a free module over the ring of additive polynomials with coefficients in $K$. Finally, we prove that every minimal purely wild extension of a henselian valued field is generated by a $p$-polynomial.

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