§ Normal field extensions
§ Normal extension
- (1) For an extension , if a polynomial and has a root has all its roots in . So splits into linear factors for .
- (2) [equivalent ] is the splitting field over of some set of polynomials.
- (3) [equivalent ] Consider . Then any automorphism of (ie, aut that fixes pointwise) maps to [fixes as a set, NOT pointwise ].
- Eq: is not normal
- Eq: is a normal extension because it's the splitting field of .
§ (1) implies (2)
- (1) We know that has a root in implies has all rots in .
- For each , we take the minimal polynomial . Then splits over , because contains a single root of ( ).
- Thus, is the splitting field for the set of polynomials .
§ (2) implies (3)
- (2) says that is the splitting field for some set of polynomials.
- An aut that fixes acts trivially on polynomials in .
- is the set of all roots of polynomials .
- Since fixes , it also cannot change the set of roots of the polynomials. Thus the set remains invariant under . ( cannot add elements into ). It can at most permute the roots of .
§ (3) implies (1)
- (3) says that any automorphism of fixes as a set.
- We wish to show that if has a root , has all roots of .
- we claim that for any root , there is an automorphism such that .
- Consider the tower of extensions and . Both and look like because is the minimal polynomial for both and .
- Thus, we can write an a function which sends .
- Now, by uniqueness of field extensions, this map extends uniquely to a map which sends . [TODO: DUBIOUS ].
- But notice that must fix (by (3)) and . Thus, , or .
- Thus, for a polynomial with root , and for any other root of , we have that .
§ Alternative argument: Splitting field of a polynomial is normal
- Let be the splitting field of . Let have a root .
- Let be another root of . We wish to show that to show that is normal.
- There is an embedding which fixes and sends to .
- See that is also a splitting field for over inside .
- But splitting fields are unique, so .
- Since , this means as desired.
§ Degree 2 elements are normal
- Let us have a degree 2 extension
- So we have some , for a root of .
- We know that for . Thus .
- Thus, the extension is normal since contains all the roots ( ) of as soon as it contained one of them.
§ Is normality of extensions transitivte?
- Consider . If is normal, is normal, then is normal?
- Answer: NO!
- Counter-example: .
- Each of the two pieces are normal since they are degree two. But the full tower is not normal, because has minimial polynomial .
- On the other hand, has a minimal polynomial .
- So, normality is not transitive!
- Another way of looking at it: We want to show that where . Since is normal, and is an autormophism of , we have [by normal ]. Since is normal, we must have . Therefore, we are done?
- NO! The problem is that is not a legal automorphism of , since fixes as a set ( ), and not pointwise ( for all .)