07-401/Class Notes for March 7 - Revision history

Drorbn at 15:38, 22 April 2007

2007-04-22T15:38:14Z

← Older revision		Revision as of 11:38, 22 April 2007
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	'''Example.''' <math>x^2-t\in{\mathbb Z}_2(t)[x]</math> is irreducible and has a single zero of multiplicity 2 within its splitting field over <math>{\mathbb Z}_2(t)[x]</math>.		'''Example.''' <math>x^2-t\in{\mathbb Z}_2(t)[x]</math> is irreducible and has a single zero of multiplicity 2 within its splitting field over <math>{\mathbb Z}_2(t)[x]</math>.

	==~~Page~~ 1==		==Lecture Notes==

			===Page 1===

	07-401 March 7 NOTES		07-401 March 7 NOTES
	[[Image:07-401 Mar.07.07 0001.jpg\|400px]]		[[Image:07-401 Mar.07.07 0001.jpg\|400px]]

	==Page 2==		===Page 2===

	07-401 March 7 NOTES		07-401 March 7 NOTES
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	==Page 3==		===Page 3===

	07-401 March 7 NOTES		07-401 March 7 NOTES
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	==Page 4==		===Page 4===

	07-401 March 7 NOTES		07-401 March 7 NOTES
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	==Page 5==		===Page 5===

	07-401 March 7 NOTES		07-401 March 7 NOTES
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	==Page 6==		===Page 6===

	07-401 March 7 NOTES		07-401 March 7 NOTES

Drorbn at 15:36, 22 April 2007

2007-04-22T15:36:59Z

← Older revision		Revision as of 11:36, 22 April 2007
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	07-401 March 7 NOTES		07-401 March 7 NOTES
	[[Image:07-401 Mar.07.07 0001.jpg]]		[[Image:07-401 Mar.07.07 0001.jpg\|400px]]

	==Page 2==		==Page 2==

	07-401 March 7 NOTES		07-401 March 7 NOTES
	[[Image:07-401 Mar.07.07 0002.jpg]]		[[Image:07-401 Mar.07.07 0002.jpg\|400px]]

	==Page 3==		==Page 3==

	07-401 March 7 NOTES		07-401 March 7 NOTES
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	07-401 March 7 NOTES		07-401 March 7 NOTES
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Drorbn: /* Zeros of Irreducible Polynomials */

2007-03-14T17:48:54Z

Zeros of Irreducible Polynomials

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

			(This section was not covered on March 7, parts of it will be covered later on).

	'''Definition.''' The derivative of a polynomial.		'''Definition.''' The derivative of a polynomial.
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	'''Example.''' <math>x^2-t\in{\mathbb Z}_2(t)[x]</math> is irreducible and has a single zero of multiplicity 2 within its splitting field over <math>{\mathbb Z}_2(t)[x]</math>.		'''Example.''' <math>x^2-t\in{\mathbb Z}_2(t)[x]</math> is irreducible and has a single zero of multiplicity 2 within its splitting field over <math>{\mathbb Z}_2(t)[x]</math>.



	==Page 1==		==Page 1==

Clkkang at 03:38, 8 March 2007

2007-03-08T03:38:47Z

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	____________________________________________________________________

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	07-401 March 7 NOTES		07-401 March 7 NOTES
	[[Image:07-401 Mar.07.07 0002.jpg ]]		[[Image:07-401 Mar.07.07 0002.jpg]]

	==Page 3==		==Page 3==

Clkkang at 03:37, 8 March 2007

2007-03-08T03:37:59Z

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	____________________________________________________________________		____________________________________________________________________

	'''Bold text'''CLASS NOTES
	==Page 1==		==Page 1==

Clkkang at 03:36, 8 March 2007

2007-03-08T03:36:57Z

← Older revision		Revision as of 23:36, 7 March 2007
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	'''Example.''' <math>x^2-t\in{\mathbb Z}_2(t)[x]</math> is irreducible and has a single zero of multiplicity 2 within its splitting field over <math>{\mathbb Z}_2(t)[x]</math>.		'''Example.''' <math>x^2-t\in{\mathbb Z}_2(t)[x]</math> is irreducible and has a single zero of multiplicity 2 within its splitting field over <math>{\mathbb Z}_2(t)[x]</math>.



			____________________________________________________________________
			'''Bold text'''CLASS NOTES
			==Page 1==

			07-401 March 7 NOTES
			[[Image:07-401 Mar.07.07 0001.jpg]]

			==Page 2==

			07-401 March 7 NOTES
			[[Image:07-401 Mar.07.07 0002.jpg ]]

			==Page 3==

			07-401 March 7 NOTES
			[[Image:07-401 Mar.07.07 0003.jpg ]]

			==Page 4==

			07-401 March 7 NOTES
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			==Page 5==

			07-401 March 7 NOTES
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			==Page 6==

			07-401 March 7 NOTES
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Sm: /* Class Plan */

2007-03-08T03:17:37Z

Class Plan

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	Some discussion of our general plan.		Some discussion of our general plan.

	Lecture ~~notes~~ [[07-401/Notes\|notes]]		Lecture [[07-401/Notes\|notes]]

	===Extension Fields===		===Extension Fields===

Sm: /* Class Plan */

2007-03-08T03:15:04Z

Class Plan

← Older revision		Revision as of 23:15, 7 March 2007
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	Some discussion of our general plan.		Some discussion of our general plan.

			Lecture notes [[07-401/Notes\|notes]]

	===Extension Fields===		===Extension Fields===

Drorbn at 21:41, 7 March 2007

2007-03-07T21:41:43Z

← Older revision		Revision as of 17:41, 7 March 2007
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	{{07-401/Navigation}}		{{07-401/Navigation}}
	{{In Preparation}}

	==Class Plan==		==Class Plan==
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	'''Definition.''' <math>F(a_1,\ldots,a_n)</math>.		'''Definition.''' <math>F(a_1,\ldots,a_n)</math>.

	'''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>.		'''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[x]/\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>.

	'''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>.
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	'''Theorem.''' Any two splitting fields for <math>f\in F[x]</math> over <math>F</math> are isomorphic.		'''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 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.		'''Lemma 2.''' Isomorphisms can be extended to splitting fields.
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	'''Theorem.''' <math>f\in F[x]</math> has a multiple zero in some extension field of <math>F</math> iff <math>f</math> and <math>f'</math> have a common factor of positive degree.		'''Theorem.''' <math>f\in F[x]</math> has a multiple zero in some extension field of <math>F</math> iff <math>f</math> and <math>f'</math> have a common factor of positive degree.

			'''Lemma.''' The property of "being relatively prime" is preserved under extensions.

	'''Theorem.''' Let <math>f\in F[x]</math> be irreducible. If <math>\operatorname{char}F=0</math>, then <math>f</math> has no multiple zeros in any extension of <math>F</math>. If <math>\operatorname{char}F=p>0</math>, then <math>f</math> has multiple zeros (in some extension) iff it is of the form <math>g(x^p)</math> for some <math>g\in F[x]</math>.		'''Theorem.''' Let <math>f\in F[x]</math> be irreducible. If <math>\operatorname{char}F=0</math>, then <math>f</math> has no multiple zeros in any extension of <math>F</math>. If <math>\operatorname{char}F=p>0</math>, then <math>f</math> has multiple zeros (in some extension) iff it is of the form <math>g(x^p)</math> for some <math>g\in F[x]</math>.

Drorbn at 20:48, 7 March 2007

2007-03-07T20:48:11Z

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 Some discussion of the [[07-401/Term Test|term test]] and [[07-401/Homework Assignment 6|HW6]].
 ===Extension Fields===
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 '''Theorem.''' <math>f\in F[x]</math> has a multiple zero in some extension field of <math>F</math> iff <math>f</math> and <math>f'</math> have a common factor of positive degree.