07-401/Class Notes for March 7
Some discussion of our general plan.
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 .
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 .
Definition. splits in , a splitting field for over .
Theorem. A splitting field always exists.
Example. over .
Example. Factor within its splitting field .
Theorem. Any two splitting fields for over are isomorphic.
Lemma 1. If irreducible over , an isomorphism, a root of (in some ), a root of in some , then .
Lemma 2. Isomorphisms can be extended to splitting fields.
Zeros of Irreducible Polynomials
(This section was not covered on March 7, parts of it will be covered later on).
Definition. The derivative of a polynomial.
Claim. The derivative operation is linear and satisfies Leibnitz's law.
Theorem. has a multiple zero in some extension field of iff and have a common factor of positive degree.
Lemma. The property of "being relatively prime" is preserved under extensions.
Theorem. Let be irreducible. If , then has no multiple zeros in any extension of . If , then has multiple zeros (in some extension) iff it is of the form for some .
Definition. A perfect field.
Theorem. A finite field is perfect.
Theorem. An irreducible polynomial over a perfect field has no multiple zeros (in any extension).
Theorem. Let be irreducible and let be the splitting field of over . Then in all zeros of have the same multiplicity.
Corollary. as above must have the form for some and .
Example. is irreducible and has a single zero of multiplicity 2 within its splitting field over .