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


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 E of F.

Theorem. For every non-constant polynomial f in F[x] there is an extension E of F in which f has a zero.

Example x^2+1 over {\mathbb R}.

Example x^5+2x^2+2x+2=(x^2+1)(x^3+2x+2) over {\mathbb Z}/3.

Definition. F(a_1,\ldots,a_n).

Theorem. If a is a root of an irreducible polynomial p\in F[x], within some extension field E of F, then F(a)\cong F[x]/\langle p\rangle, and \{1,a,a^2,\ldots,a^{n-1}\} (here n=\deg p) is a basis for F(a) over F.

Corollary. In this case, F(a) depends only on p.

Splitting Fields

Definition. f\in F[x] splits in E/F, a splitting field for f over F.

Theorem. A splitting field always exists.

Example. x^4-x^2-2=(x^2-2)(x^2+1) over {\mathbb Q}.

Example. Factor x^2+x+2\in{\mathbb Z}_3[x] within its splitting field {\mathbb Z}_3[x]/\langle x^2+x+2\rangle.

Theorem. Any two splitting fields for f\in F[x] over F are isomorphic.

Lemma 1. If p\in F[x] irreducible over F, \phi:F\to F' an isomorphism, a a root of p (in some E/F), a' a root of \phi(p) in some E'/F', then F(a)\cong F'(a').

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. f\in F[x] has a multiple zero in some extension field of F iff f and f' have a common factor of positive degree.

Lemma. The property of "being relatively prime" is preserved under extensions.

Theorem. Let f\in F[x] be irreducible. If \operatorname{char}F=0, then f has no multiple zeros in any extension of F. If \operatorname{char}F=p>0, then f has multiple zeros (in some extension) iff it is of the form g(x^p) for some g\in F[x].

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 f\in F[x] be irreducible and let E be the splitting field of f over F. Then in E all zeros of f have the same multiplicity.

Corollary. f as above must have the form a(x-a_1)^n\cdots(x-a_k)^n for some a\in F and a_1,\ldots,a_k\in E.

Example. x^2-t\in{\mathbb Z}_2(t)[x] is irreducible and has a single zero of multiplicity 2 within its splitting field over {\mathbb Z}_2(t)[x].


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