07-401/Class Notes for March 7: Difference between revisions
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{{07-401/Navigation}} |
{{07-401/Navigation}} |
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{{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=== |
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Revision as of 14:53, 7 March 2007
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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.
![{\displaystyle F[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39bc9f9d8679fc385df3bccf9694283b796f3216)
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 .
![{\displaystyle p\in F[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05a3455e0a007d78c63b1dc8e435af11a6b61642)
![{\displaystyle F(a)\cong F[a]/\langle p\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1805e69905ed552abaf1abbb518f7ef7046a1e5)
Corollary. In this case, depends only on .
Corollary. If irreducible over , an isomorphism, a root of (in some ), a root of in some , then .
![{\displaystyle F[a]\cong F'[a']}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7940099f27f9e55b46f651fd4ca5fbc49ec88049)
Theorem. A splitting field always exists.
Example. over .
Splitting Fields
Definition. splits in , a splitting field for over .
![{\displaystyle f\in F[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ccbe5f3f86529d2b1b617a852f0368e16bba2d0)
