Commutative algebra homework
Last update: 02 March 2012
Let be a (commutative?) topological group with a fundamental system of neighborhoods of 0 which are subgroups,
Show that the completion of
by showing that a sequence in is coherent if and only if it is Cauchy.
Let be a set and let
- Define relation on , equivalence relation on , and equivalence class and give some illustrative examples.
- Define partition of and partition of and give some illustrative examples.
- State and prove a theorem which makes precise the concept that equivalence relations and partitions are interchangable.
Define the notion of a function
as a subset of .
Consider the graph of and write down precise conditions for a subset of to be the graph of a function.
- Define isomorphism of sets, and is bijective and give some illustrative examples.
- Prove that a function is an isomorphism of sets if and only if is bijective.
Let and be topological spaces. Let and let be a function.
- Define is continuous at and give some illustrative examples.
and give some illustrative examples.
- Prove the following fundamental theorem:
Week 1 Problems
Lecture 2 - 29/02/2012
Expand as a power series beginning with
Lecture 3 - 02/03/2012
Do the exercises on the p-adic numbers page.
Using the definition of the ring of fractions with denominators in from the lectures, prove that is an equivalence relation, that
is well defined, that
is well defined, and with and is a ring.
Week 2 Problems
Lecture 2 - 07/03/2012
is an module homomorphism.
Show that is a functor (i.e.
Lecture 3 - 09/03/2012
Let be a linear transformation from a vector space into an algebra Suppose that if then Show that if then
Compute the dimension of
Determine (make precise) the definition of
as used in the construction of
Week 3 Problems
Lecture 1 - 12/03/2012
Show that if is a cyclic group then or for some
Show that the group can be presented by generators with relations
Notes and References