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

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


===Splitting Fields===
===Splitting Fields===
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'''Example.''' Factor <math>x^2+x+2\in{\mathbb Z}_3[x]</math> within its splitting field <math>{\mathbb Z}_3[x]/\langle x^2+x+2\rangle</math>.
'''Example.''' Factor <math>x^2+x+2\in{\mathbb Z}_3[x]</math> within its splitting field <math>{\mathbb Z}_3[x]/\langle x^2+x+2\rangle</math>.

'''Theorem.''' Any two splitting fields for <math>f\in F[x]</math> over <math>F</math> are isomorphic.

⚫
'''Lemma 1.''' 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>.

'''Lemma 2.''' Isomorphisms can be extended to splitting fields.


===Zeros of Irreducible Polynomials===
===Zeros of Irreducible Polynomials===

Revision as of 14:10, 7 March 2007

07-401/Navigation Panel  
# Week of... Links
1 Jan 10 About, Notes, HW1
2 Jan 17 HW2, Notes
3 Jan 24 HW3, Photo, Notes
4 Jan 31 HW4, Notes
5 Feb 7 HW5, Notes
6 Feb 14 On TT, Notes
R Feb 21 Reading week
7 Feb 28 Term Test
8 Mar 7 HW6, Notes
9 Mar 14 HW7, Notes
10 Mar 21 HW8, E8, Notes
11 Mar 28 HW9, Notes
12 Apr 4 HW10, Notes
13 Apr 11 Notes, PM
S Apr 16-20 Study Period
F Apr 24 Final
07-401 Class Photo.jpg
Add your name / see who's in!
Register of Good Deeds
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 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} .

Theorem. For every non-constant 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 f} in there is an extension 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} in which 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} has a zero.

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^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 R}} .

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^5+2x^2+2x+2=(x^2+1)(x^3+2x+2)} 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 Z}/3} .

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(a_1,\ldots,a_n)} .

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, 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)} 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} .

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} .

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}} .

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

Theorem. Any two splitting fields 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\in F[x]} 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} are isomorphic.

Lemma 1. 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} ), 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 \phi(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'} , 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']} .

Lemma 2. Isomorphisms can be extended to splitting fields.

Zeros of Irreducible Polynomials

Perfect Fields

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