07-401/Class Notes for March 7
From Drorbn
|
In Preparation
The information below is preliminary and cannot be trusted! (v)
Contents |
Class Plan
Some discussion of the term test and HW6.
Extension Fields
Definition. An extension field of .
Theorem. For every non-constant polynomial in there is an extension of in which has a zero.
Example over .
Example over .
Definition. .
Theorem. If is a root of an irreducible polynomial , within some extension field of , then , and (here ) is a basis for over .
Corollary. In this case, depends only on .
Corollary. If irreducible over , an isomorphism, a root of (in some ), a root of in some , then .
Splitting Fields
Definition. splits in , a splitting field for over .
Theorem. A splitting field always exists.
Example. over .
Example. Factor within its splitting field .