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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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'''Example.''' <math>x^4-x^2-2=(x^2-2)(x^2+1)</math> over <math>{\mathbb Q}</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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===Zeros of Irreducible Polynomials=== |
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===Zeros of Irreducible Polynomials=== |
Revision as of 14:06, 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 .
Corollary. If irreducible over , an isomorphism, a root of (in some ), a root of in some , then .
Splitting Fields
Definition. splits in , a splitting field for over .
Theorem. A splitting field always exists.
Example. over .
Example. Factor within its splitting field .
Zeros of Irreducible Polynomials
Perfect Fields