07-401/Class Notes for April 11 - Revision history

Drorbn at 18:53, 2 April 2008

2008-04-02T18:53:58Z

← Older revision		Revision as of 13:53, 2 April 2008
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	{{Equation*\|<math>\Psi:\quad H\mapsto E_H</math>.}}		{{Equation*\|<math>\Psi:\quad H\mapsto E_H</math>.}}
	This correspondence has the following further properties:		This correspondence has the following further properties:
	# It is inclusion-reversing: if <math>H_1\subset H_2</math> then <math>E_{H_1}\supset E_{H_2}</math> and if <math>K_1\subset K_2</math> then <math>\operatorname{Gal}(E/K_1)>\operatorname{Gal}(E/~~K_1~~)</math>.		# It is inclusion-reversing: if <math>H_1\subset H_2</math> then <math>E_{H_1}\supset E_{H_2}</math> and if <math>K_1\subset K_2</math> then <math>\operatorname{Gal}(E/K_1)>\operatorname{Gal}(E/K_2)</math>.
	# It is degree/index respecting: <math>[E:K]=\|\operatorname{Gal}(E/K)\|</math> and <math>[K:F]=[\operatorname{Gal}(E/F):\operatorname{Gal}(E/K)]</math>.		# It is degree/index respecting: <math>[E:K]=\|\operatorname{Gal}(E/K)\|</math> and <math>[K:F]=[\operatorname{Gal}(E/F):\operatorname{Gal}(E/K)]</math>.
	# Splitting fields correspond to normal subgroups: If <math>K</math> in <math>E/K/F</math> is the splitting field of a polynomial in <math>F[x]</math> then <math>\operatorname{Gal}(E/K)</math> is normal in <math>\operatorname{Gal}(E/F)</math> and <math>\operatorname{Gal}(K/F)\cong\operatorname{Gal}(E/F)/\operatorname{Gal}(E/K)</math>.		# Splitting fields correspond to normal subgroups: If <math>K</math> in <math>E/K/F</math> is the splitting field of a polynomial in <math>F[x]</math> then <math>\operatorname{Gal}(E/K)</math> is normal in <math>\operatorname{Gal}(E/F)</math> and <math>\operatorname{Gal}(K/F)\cong\operatorname{Gal}(E/F)/\operatorname{Gal}(E/K)</math>.

Drorbn: /* The Fundamental Theorem of Galois Theory */

2008-03-30T21:54:19Z

The Fundamental Theorem of Galois Theory

← Older revision		Revision as of 16:54, 30 March 2008
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	==The Fundamental Theorem of Galois Theory==		==The Fundamental Theorem of Galois Theory==

			'''Note.''' An updated version of this section is at [[08-401/The Fundamental Theorem]].

	It seems we will not have time to prove the Fundamental Theorem of Galois Theory in full. Thus this note is about what we will be missing. The statement appearing here, which is a weak version of the full theorem, is taken from Gallian's book and is meant to match our discussion in class. The proof is taken from Hungerford's book, except modified to fit our notations and conventions and simplified as per our weakened requirements.		It seems we will not have time to prove the Fundamental Theorem of Galois Theory in full. Thus this note is about what we will be missing. The statement appearing here, which is a weak version of the full theorem, is taken from Gallian's book and is meant to match our discussion in class. The proof is taken from Hungerford's book, except modified to fit our notations and conventions and simplified as per our weakened requirements.

Drorbn: /* Extra Practice Problems with Solutions */

2007-04-22T16:17:09Z

Extra Practice Problems with Solutions

← Older revision		Revision as of 11:17, 22 April 2007
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	==Extra Practice Problems with Solutions==		==Extra Practice Problems with Solutions==

	Attached were some problems [[User:Clkkang]] found on the net with solutions. They may be very useful when studying for problems involving galois theory and other subjects. ~~Most~~ were deleted by {{Dror}} for fear of copyright violations.		Attached were some problems [[User:Clkkang]] found on the net with solutions. They may be very useful when studying for problems involving galois theory and other subjects. They were deleted by {{Dror}} for fear of copyright violations (it will be appropriate for [[User:Clkkang]] to post here links to the original sources of those pages).

	The following attachments are notes that [[User:Clkkang]] had taken from the text, it "MAY BE" a little easier to read than the text due to the formatting:		The following attachments are notes that [[User:Clkkang]] had taken from the text, it "MAY BE" a little easier to read than the text due to the formatting:

Drorbn at 16:08, 22 April 2007

2007-04-22T16:08:46Z

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 Clearly, <math>\rho</math> is a group homomorphism. The kernel of <math>\rho</math> is those automorphisms of <math>E</math> whose restriction to <math>K</math> is the identity. That is, it is <math>\operatorname{Gal}(E/K)</math>. Finally, as <math>E/F</math> is a splitting extension, so is <math>E/K</math>. So every automorphism of <math>K</math> extends to an automorphism of <math>E</math> by the uniqueness statement for splitting extensions. But this means that <math>\rho</math> is onto. <math>\Box</math>
 ==Extra Practice Problems with Solutions==
 [[Image:07-401 Ch 32 Notes Part 1.pdf]]
-==Page 2==
+===Page 2===
-==Page 3==
+===Page 3===
-==Page 4==
+===Page 4===

Clkkang at 09:22, 14 April 2007

2007-04-14T09:22:11Z

← Older revision		Revision as of 04:22, 14 April 2007
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	'''~~Notes~~ ~~From~~ ~~The~~ ~~Text~~ (MAY BE A little ~~bit~~ easier to read ~~then~~ text?)'''		'''The following attachments are notes that I've taken from the text, it "MAY BE" a little easier to read than the text due to the formatting'''

	[[Image:07-401 Ch 32 Notes Part 1.pdf]]		[[Image:07-401 Ch 32 Notes Part 1.pdf]]

Clkkang at 09:15, 14 April 2007

2007-04-14T09:15:13Z

← Older revision		Revision as of 04:15, 14 April 2007
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	[[Image:07-401 Min. poly, Ext., Proving Algebraic.pdf]]		[[Image:07-401 Min. poly, Ext., Proving Algebraic.pdf]]


	'''My ~~Text~~ ~~Book~~ ~~Notes~~ (MAY BE A little bit easier to read then text?)'''		'''Notes From The Text (MAY BE A little bit easier to read then text?)'''

			[[Image:07-401 Ch 32 Notes Part 1.pdf]]

74.99.212.47 at 09:11, 14 April 2007

2007-04-14T09:11:40Z

← Older revision		Revision as of 04:11, 14 April 2007
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			==Extra Practice Problems with Solutions==
	==EXTRA PRACTICE PROBLEMS WITH SOLUTIONS==
	'''Attached are some problems I've found on the net with solutions. I think they may be very useful when studying for problems involving galois theory and other subjects.'''		'''Attached are some problems I've found on the net with solutions. I think they may be very useful when studying for problems involving galois theory and other subjects.'''

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	[[Image:07-401 Min. poly, Ext., Proving Algebraic.pdf]]		[[Image:07-401 Min. poly, Ext., Proving Algebraic.pdf]]

			'''My Text Book Notes (MAY BE A little bit easier to read then text?)'''

Clkkang at 01:16, 12 April 2007

2007-04-12T01:16:33Z

← Older revision		Revision as of 20:16, 11 April 2007
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	Clearly, <math>\rho</math> is a group homomorphism. The kernel of <math>\rho</math> is those automorphisms of <math>E</math> whose restriction to <math>K</math> is the identity. That is, it is <math>\operatorname{Gal}(E/K)</math>. Finally, as <math>E/F</math> is a splitting extension, so is <math>E/K</math>. So every automorphism of <math>K</math> extends to an automorphism of <math>E</math> by the uniqueness statement for splitting extensions. But this means that <math>\rho</math> is onto. <math>\Box</math>		Clearly, <math>\rho</math> is a group homomorphism. The kernel of <math>\rho</math> is those automorphisms of <math>E</math> whose restriction to <math>K</math> is the identity. That is, it is <math>\operatorname{Gal}(E/K)</math>. Finally, as <math>E/F</math> is a splitting extension, so is <math>E/K</math>. So every automorphism of <math>K</math> extends to an automorphism of <math>E</math> by the uniqueness statement for splitting extensions. But this means that <math>\rho</math> is onto. <math>\Box</math>



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	[[Image:07-401 Min. poly, Ext., Proving Algebraic.pdf]]		[[Image:07-401 Min. poly, Ext., Proving Algebraic.pdf]]

Theo at 01:07, 12 April 2007

2007-04-12T01:07:09Z

← Older revision		Revision as of 20:07, 11 April 2007
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	07-401 Apr/04/11 LECTURE NOTES		07-401 Apr/04/11 LECTURE NOTES
	[[Image:07-401 lecture 13 pg 1.jpg]]		[[Image:07-401 lecture 13 pg 1.jpg]]

			==Page 2==

			07-401 Apr/04/11 LECTURE NOTES
			[[Image:07-401 lecture 13 pg 2.jpg]]

			==Page 3==

			07-401 Apr/04/11 LECTURE NOTES
			[[Image:07-401 lecture 13 pg 3.jpg]]

			==Page 4==

			07-401 Apr/04/11 LECTURE NOTES
			[[Image:07-401 lecture 13 pg 4.jpg]]

Clkkang at 01:03, 12 April 2007

2007-04-12T01:03:16Z

← Older revision		Revision as of 20:03, 11 April 2007
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	'''Extra Practice with Polynomials, Irreduciblity ~~and Galois~~:'''		'''A Little Less Galois, Extra Practice with Polynomials, Irreduciblity:'''

	[[Image:07-401 minimal polynomial, Extensions.pdf]]		[[Image:07-401 minimal polynomial, Extensions.pdf]]
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	[[Image:07-401 Extensions and Galois Soln.pdf]]		[[Image:07-401 Extensions and Galois Soln.pdf]]

			[[Image:07-401 Min. poly, Ext., Proving Algebraic.pdf]]



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