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

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==Class Plan==

Some discussion of the [[07-401/Term Test|term test]] and [[07-401/Homework Assignment 6|HW6]].

===Extension Fields===

'''Definition.''' An extension field <math>E</math> of <math>F</math>.

'''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.

'''Example''' <math>x^2+1</math> over <math>{\mathbb R}</math>.

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

'''Definition.''' <math>F(a_1,\ldots,a_n)</math>.

'''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>.

'''Corollary.''' In this case, <math>F(a)</math> depends only on <math>p</math>.

'''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>.

'''Theorem.''' A splitting field always exists.

'''Example.''' <math>x^4-x^2-2=(x^2-2)(x^2+1)</math> over <math>{\mathbb Q}</math>.

===Splitting Fields===

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

===Zeros of Irreducible Polynomials===

===Perfect Fields===

Revision as of 13:53, 7 March 2007

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# Week of... Links
1 Jan 10 About, Notes, HW1
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4 Jan 31 HW4, Notes
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6 Feb 14 On TT, Notes
R Feb 21 Reading week
7 Feb 28 Term Test
8 Mar 7 HW6, Notes
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S Apr 16-20 Study Period
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In Preparation

The information below is preliminary and cannot be trusted! (v)

Class Plan

Some discussion of the term test and HW6.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in F[x]} , within some extension field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(a)\cong F[a]/\langle p\rangle} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1,a,a^2,\ldots,a^{n-1}\}} (here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=\deg p} ) is a basis for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(a)} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} .

Corollary. In this case, depends only on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} .

Corollary. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in F[x]} irreducible over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi:F\to F'} an isomorphism, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} a root of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} (in some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E/F} ), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a'} a root of in some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E'/F'} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F[a]\cong F'[a']} .

Theorem. A splitting field always exists.

Example. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^4-x^2-2=(x^2-2)(x^2+1)} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Q}} .

Splitting Fields

Definition. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\in F[x]} splits in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E/F} , a splitting field for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} .

Zeros of Irreducible Polynomials

Perfect Fields

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