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'''Corollary.''' In this case, <math>F(a)</math> depends only on <math>p</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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===Splitting Fields=== |
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===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>. |
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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>. |
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'''Theorem.''' Any two splitting fields for <math>f\in F[x]</math> over <math>F</math> are isomorphic. |
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''' 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>. |
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'''Lemma 2.''' Isomorphisms can be extended to splitting fields. |
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===Zeros of Irreducible Polynomials=== |
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===Zeros of Irreducible Polynomials=== |
Revision as of 14:10, 7 March 2007
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Week of...
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Links
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1
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Jan 10
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About, Notes, HW1
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2
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Jan 17
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HW2, Notes
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3
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Jan 24
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HW3, Photo, Notes
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4
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Jan 31
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HW4, Notes
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5
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Feb 7
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HW5, Notes
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6
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Feb 14
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On TT, Notes
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R
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Feb 21
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Reading week
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7
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Feb 28
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Term Test
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8
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Mar 7
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HW6, Notes
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9
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Mar 14
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HW7, Notes
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10
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Mar 21
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HW8, E8, Notes
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11
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Mar 28
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HW9, Notes
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12
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Apr 4
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HW10, Notes
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13
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Apr 11
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Notes, PM
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S
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Apr 16-20
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Study Period
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F
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Apr 24
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Final
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Add your name / see who's in!
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Register of Good Deeds
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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 , 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
Perfect Fields