07-401/Class Notes for March 7: Difference between revisions
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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 13:53, 7 March 2007
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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 [math]\displaystyle{ E }[/math] of [math]\displaystyle{ F }[/math].
Theorem. For every non-constant polynomial [math]\displaystyle{ f }[/math] in [math]\displaystyle{ F[x] }[/math] there is an extension [math]\displaystyle{ E }[/math] of [math]\displaystyle{ F }[/math] in which [math]\displaystyle{ f }[/math] has a zero.
Example [math]\displaystyle{ x^2+1 }[/math] over [math]\displaystyle{ {\mathbb R} }[/math].
Example [math]\displaystyle{ x^5+2x^2+2x+2=(x^2+1)(x^3+2x+2) }[/math] over [math]\displaystyle{ {\mathbb Z}/3 }[/math].
Definition. [math]\displaystyle{ F(a_1,\ldots,a_n) }[/math].
Theorem. If [math]\displaystyle{ a }[/math] is a root of an irreducible polynomial [math]\displaystyle{ p\in F[x] }[/math], within some extension field [math]\displaystyle{ E }[/math] of [math]\displaystyle{ F }[/math], then [math]\displaystyle{ F(a)\cong F[a]/\langle p\rangle }[/math], and [math]\displaystyle{ \{1,a,a^2,\ldots,a^{n-1}\} }[/math] (here [math]\displaystyle{ n=\deg p }[/math]) is a basis for [math]\displaystyle{ F(a) }[/math] over [math]\displaystyle{ F }[/math].
Corollary. In this case, [math]\displaystyle{ F(a) }[/math] depends only on [math]\displaystyle{ p }[/math].
Corollary. If [math]\displaystyle{ p\in F[x] }[/math] irreducible over [math]\displaystyle{ F }[/math], [math]\displaystyle{ \phi:F\to F' }[/math] an isomorphism, [math]\displaystyle{ a }[/math] a root of [math]\displaystyle{ p }[/math] (in some [math]\displaystyle{ E/F }[/math]), [math]\displaystyle{ a' }[/math] a root of [math]\displaystyle{ \phi(p) }[/math] in some [math]\displaystyle{ E'/F' }[/math], then [math]\displaystyle{ F[a]\cong F'[a'] }[/math].
Theorem. A splitting field always exists.
Example. [math]\displaystyle{ x^4-x^2-2=(x^2-2)(x^2+1) }[/math] over [math]\displaystyle{ {\mathbb Q} }[/math].
Splitting Fields
Definition. [math]\displaystyle{ f\in F[x] }[/math] splits in [math]\displaystyle{ E/F }[/math], a splitting field for [math]\displaystyle{ f }[/math] over [math]\displaystyle{ F }[/math].
