14-1100/Homework Assignment 4 - Revision history

Drorbn at 17:21, 6 November 2014

2014-11-06T17:21:57Z

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	{{In Preparation}}

	This assignment is due at class time on Thursday, November 20, 2011.		This assignment is due at class time on Thursday, November 20, 2011.

Drorbn at 14:51, 6 November 2014

2014-11-06T14:51:58Z

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	===Solve the following questions===		===Solve the following questions===

			'''Problem 1.''' (Klein's 1983 course)
	'''Problem 1.''' Prove that a ring <math>R</math> is a PID iff it is a UFD in which <math>\gcd(a,b)\in\langle a, b\rangle</math> for every non-zero <math>a,b\in R</math>.
			# Show that the ideal <math>I=\langle 3,\, x^3-x^2+2x-1\rangle</math> inside the ring <math>{\mathbb Z}[x]</math> is not principal.
			# Is <math>{\mathbb Z}[x]/I</math> a domain?

	'''Problem 2.''' ~~(Lang) Show~~ that ~~the~~ ring <math>~~{\mathbb Z}[i]=\{a+ib\colon a,b\in{\mathbb Z}\}\subset{\mathbb C}~~</math> is a PID ~~and~~ ~~hence~~ a UFD. ~~What~~ ~~are~~ ~~the~~ ~~units~~ of ~~that~~ ~~ring?~~		'''Problem 2.''' Prove that a ring <math>R</math> is a PID iff it is a UFD in which <math>\gcd(a,b)\in\langle a, b\rangle</math> for every non-zero <math>a,b\in R</math>.

	'''Problem 3.''' (~~Dummit~~ ~~and~~ ~~Foote)~~ In <math>{\mathbb Z}[i]~~</math>~~, ~~find the greatest common divisor of~~ ~~<math>85~~</math> ~~and~~ ~~<math>1+13i</math>,~~ and ~~express~~ it as a ~~linear~~ ~~combination~~ of ~~these~~ ~~two elements.~~		'''Problem 3.''' (Lang) Show that the ring <math>{\mathbb Z}[i]=\{a+ib\colon a,b\in{\mathbb Z}\}\subset{\mathbb C}</math> is a PID and hence a UFD. What are the units of that ring?

	'''Problem 4.''' (~~Hard!)~~ ~~Show~~ ~~that~~ ~~the quotient ring~~ <math>{\mathbb Q}[~~x,y~~]/~~\langle~~ ~~x^2+y^2-~~1~~\rangle~~</math> is ~~not~~ a ~~UFD~~.		'''Problem 4.''' (Dummit and Foote) In <math>{\mathbb Z}[i]</math>, find the greatest common divisor of <math>85</math> and <math>1+13i</math>, and express it as a linear combination of these two elements.

			'''Problem 5.''' (Klein's 1983 course) Show that <math>{\mathbb Z}[\sqrt{10}]</math> is not a UFD.

			'''Problem 6.''' (Hard!) Show that the quotient ring <math>{\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle</math> is not a UFD.

Drorbn: Created page with "{{14-1100/Navigation}} {{In Preparation}} This assignment is due at class time on Thursday, November 20, 2011. ===Solve the following questions=== '''Problem 1.''' Prove th..."

2014-11-06T14:35:25Z

Created page with "{{14-1100/Navigation}} {{In Preparation}} This assignment is due at class time on Thursday, November 20, 2011. ===Solve the following questions=== '''Problem 1.''' Prove th..."

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{{14-1100/Navigation}}
{{In Preparation}}

This assignment is due at class time on Thursday, November 20, 2011.

===Solve the following questions===

'''Problem 1.''' Prove that a ring <math>R</math> is a PID iff it is a UFD in which <math>\gcd(a,b)\in\langle a, b\rangle</math> for every non-zero <math>a,b\in R</math>.

'''Problem 2.''' (Lang) Show that the ring <math>{\mathbb Z}[i]=\{a+ib\colon a,b\in{\mathbb Z}\}\subset{\mathbb C}</math> is a PID and hence a UFD. What are the units of that ring?

'''Problem 3.''' (Dummit and Foote) In <math>{\mathbb Z}[i]</math>, find the greatest common divisor of <math>85</math> and <math>1+13i</math>, and express it as a linear combination of these two elements.

'''Problem 4.''' (Hard!) Show that the quotient ring <math>{\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle</math> is not a UFD.