Difference between revisions of "07-401/Class Notes for March 7"

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{{07-401/Navigation}}
 
{{07-401/Navigation}}
 
{{In Preparation}}
 
{{In Preparation}}
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==Class Plan==
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Some discussion of the [[07-401/Term Test|term test]] and [[07-401/Homework Assignment 6|HW6]].
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===Extension Fields===
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'''Definition.''' An extension field <math>E</math> of <math>F</math>.
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'''Theorem.''' For every non-constant polynomial <math>f</math> in <math>F[x]</math> there is an extension <math>E</math> of <math>F</math> in which <math>f</math> has a zero.
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'''Example''' <math>x^2+1</math> over <math>{\mathbb R}</math>.
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'''Example''' <math>x^5+2x^2+2x+2=(x^2+1)(x^3+2x+2)</math> over <math>{\mathbb Z}/3</math>.
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'''Definition.''' <math>F(a_1,\ldots,a_n)</math>.
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'''Theorem.''' If <math>a</math> is a root of an irreducible polynomial <math>p\in F[x]</math>, within some extension field <math>E</math> of <math>F</math>, then <math>F(a)\cong F[a]/\langle p\rangle</math>, and <math>\{1,a,a^2,\ldots,a^{n-1}\}</math> (here <math>n=\deg p</math>) is a basis for <math>F(a)</math> over <math>F</math>.
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'''Corollary.''' In this case, <math>F(a)</math> depends only on <math>p</math>.
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'''Corollary.''' If <math>p\in F[x]</math> irreducible over <math>F</math>, <math>\phi:F\to F'</math> an isomorphism, <math>a</math> a root of <math>p</math> (in some <math>E/F</math>), <math>a'</math> a root of <math>\phi(p)</math> in some <math>E'/F'</math>, then <math>F[a]\cong F'[a']</math>.
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'''Theorem.''' A splitting field always exists.
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'''Example.''' <math>x^4-x^2-2=(x^2-2)(x^2+1)</math> over <math>{\mathbb Q}</math>.
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===Splitting Fields===
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'''Definition.''' <math>f\in F[x]</math> splits in <math>E/F</math>, a splitting field for <math>f</math> over <math>F</math>.
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===Zeros of Irreducible Polynomials===
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===Perfect Fields===

Revision as of 14:53, 7 March 2007

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 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[a]/\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.

Corollary. 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'].

Theorem. A splitting field always exists.

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

Splitting Fields

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

Zeros of Irreducible Polynomials

Perfect Fields