September 17, 2011 32 Comments
The following objects are really important in mathematics:
- Closure: If and are in the field, so are and
- and are associative. That is, and
- , and
- There is an “additive identity” element such that
- For each there is a unique “additive inverse” such that
- There is a “multiplicative identity” such that
- For each (other than ) there is a unique “multiplicative inverse” such that . (i.e., don’t divide by zero!)
- EDIT: Also,
- , the set of integers is not a field. There is no integer for which , so has no multiplicative inverse.
- We can define subtraction and division to be what they should be means , where is the additive inverse of . where is the multiplicative inverse of .
- Finite sums/products only. Sorry, I don’t make the rules.
- Can you prove that for every in an arbitrary field?