# Model Theory of Fields (WS 2015/16)

**Lectures:** Tuesday 12-14, SR 403

**Exercise sessions:** Thursday 12-14, SR 403

## Lectures

- ACF, ACF_p, Quantifier elimination.
- Strong minimality of ACF_p, basic examples of strongly minimal theories, Zariski closed sets, geometric interpretation of QE, Nullstellensatz.
- ACF_p: acl and dcl, definable functions are definably piecewise rational, dimension: RM X, acl-dim X, dim V, trd_K K(V).
- ACF_p: Elimination of imaginaries.
- ACF_p: Non local modularity.
- Derivations, extension lemmas, DCF_0.
- DCF_0: Quantifier elimination.
- DCF_0: Geometric Axioms, linear differential equations.
- DCF_0: Differential algebra.
- DCF_0: Differential algebra.
- DCF_0: 1-types.
- DCF_0: Morley rank.
- DCF_0: acl and non-forking independence.
- Differential Galois theory for Picard-Vessiot extensions.
- Differential Galois theory for Picard-Vessiot extensions.

## Exercises

- Week 1
- Week 2
- Week 3
- Week 4
- Week 5
- Week 6
- Week 7
- Week 8 (No exercises, extra lecture.)
- Week 9
- Week 10: DCF_0 is not finitely axiomatizable./a>
- Week 11: Differential Nullstellensatz, isolated 1-types.
- Week 12: More on isolated 1-types.
- Week 13: DCF_0 eliminates imaginaries.
- Week 14: Non-forking independence.
- Week 15 (No exercises, extra lecture.)

## References

- [K] I. Kaplansky,
*Introduction to differential algebra*, 1957.
- [M.mtf] D. Marker,
*An Introduction to the Model Theory of Fields*.
- [M.dcf] D. Marker,
*Model Theory of Differential Fields*.
- [P.acf] A. Pillay,
*Model Theory of algebraically closed fields*, in E. Bouscaren (ed.) *Model Theory and Algebraic Geometry*, LNM 1696, 1999.
- [TZ] K. Tent und M. Ziegler,
*A course in Model Theory*, Cambridge University Press, 2012.
- [Z.dcf] M. Ziegler,
*AnfĂ¤nge der Differentialalgebra*.