University of Wisconsin-Madison |
Math 541 |
Fall 2007
|
Define monoid without identity, monoid, group, ring without identity, ring, division ring and field, and give examples. Make sure that your example of a monoid without identity is not a monoid, that your example of a monoid is not a group, etc.
Give and example of an operation on that is not associative.
Let be a group. Show that the identity element of is unique.
Let be a group and let . Show that the inverse of is unique.
Why isn't a group?
Show that .
Show that .
Show that .
Show that .
Define and prove that it is a field.
Define the quaternions and show that they are a division ring and not a field.
Define and prove that it is a group.
Define and prove that it is a ring.
For which positive integers is a field?
Let be a positive integer. An th root of unity is a complex number such that .
Let be the set of th roots of unity in . Determine , , , and graph these sets.
Let be the set of th roots of unity in . Show that is a group.
Define and prove that it is a ring.
Define and prove that it is a ring.
Define and prove that it is a field.
Define and prove that it is a ring.
Define and prove that it is a field.
Show that each element of has a unique expression in the form , where and has nonzero constant term.
Show that there exists a field with 4 elements.