Projects and Assignments in Pure Mathematics
Zorn's Lemma at Work
Zorn's Lemma (ZL), an equivalent of the Axiom of Choice, is often used to
give in full generality an existence proof for a certain mathematical
object.
For example, to prove that an arbitrary vector space has a basis
one needs to use ZL unless the vector space under consideration is
finitely generated.
This project's objective is to study a selection of
typical applications of ZL in
linear algebra (see above),
group theory
(maximal p-subgroups),
field theory (transcendence bases, algebraic
closures),
commutative algebra (maximal submodules, minimal prime ideals).
Uses of ZL within other parts of pure mathematics may also be considered.
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References
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As a first reading one may consult the parts on ZL of
Irving Kaplansky's "Set
Theory and Metric Spaces", which can be found e.g. in Section 2.3, on page
53, in Section 3.3., and in Appendix 2.
-
Thomas J. Jech, "The Axiom of Choice", Chapter 2. Now available also as a
Dover reprint.