Difference between revisions of "07-401/Class Notes for March 7"
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'''Example.''' <math>x^2-t\in{\mathbb Z}_2(t)[x]</math> is irreducible and has a single zero of multiplicity 2 within its splitting field over <math>{\mathbb Z}_2(t)[x]</math>. | '''Example.''' <math>x^2-t\in{\mathbb Z}_2(t)[x]</math> is irreducible and has a single zero of multiplicity 2 within its splitting field over <math>{\mathbb Z}_2(t)[x]</math>. | ||
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Revision as of 23:36, 7 March 2007
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Contents |
Class Plan
Some discussion of the term test and HW6.
Some discussion of our general plan.
Lecture notes
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 .
Splitting Fields
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
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 .
____________________________________________________________________ Bold textCLASS NOTES