12-240/Fields' Further proof - Revision history

Michael.Wang at 20:13, 16 September 2012

2012-09-16T20:13:15Z

← Older revision		Revision as of 16:13, 16 September 2012
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	4 0 . . . ......................		4 0 . . . ......................

	. 0 . . . ......................		.. 0 . . . ......................

	a 0 (a) (2a) (3a).........(ab)....(n-1)a ''' ( in this row, every element mod n)'''		a 0 (a) (2a) (3a).........(ab)....(n-1)a ''' ( in this row, every element mod n)'''

Michael.Wang at 20:11, 16 September 2012

2012-09-16T20:11:06Z

← Older revision		Revision as of 16:11, 16 September 2012
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	~~<nowiki>~~[[12-240]][[Image:12-240-Splash.png]]		[[12-240]][[Image:12-240-Splash.png]]

	[[12-240/Classnotes for Tuesday September 11]]		[[12-240/Classnotes for Tuesday September 11]]
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	I am still working on it.		I am still working on it.
	----Michael.Wang		----Michael.Wang
	<nowiki></nowiki>

Michael.Wang at 20:09, 16 September 2012

2012-09-16T20:09:20Z

← Older revision		Revision as of 16:09, 16 September 2012
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	[[12-240]][[Image:12-240-Splash.png]]		<nowiki>[[12-240]][[Image:12-240-Splash.png]]

	[[12-240/Classnotes for Tuesday September 11]]		[[12-240/Classnotes for Tuesday September 11]]
Line 19:		Line 19:
	IN defination of multiplication		IN defination of multiplication

	* 0 1 2 3 .......... b.......n-1		* 0 1 2 3 .......... b.......n-1

	0 0 0 0 0 ...........0.........0		0 0 0 0 0 ...........0........0

	1 0 . . . ......................		1 0 . . . ......................

	2 0 . . . ......................		2 0 . . . ......................

	3 0 . . . .......................		3 0 . . . ......................

	4 0 . . . .......................		4 0 . . . ......................

	.. 0 . . . .......................		. 0 . . . ......................

	a 0 (a) (2a) (3a)...........(ab).....(n-1)a ''' ( in this row, every element mod n)'''		a 0 (a) (2a) (3a).........(ab)....(n-1)a ''' ( in this row, every element mod n)'''

	....................................		....0.....................................

	....................................		....0.....................................

	......................................		....0.....................................

	.....................................		....0.....................................

	(n-1) 0................................		(n-1) 0...................................


	see the (a+1)th row		see the (a+1)th row

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	I am still working on it.		I am still working on it.
	----Michael.Wang		----Michael.Wang
			<nowiki></nowiki>

Michael.Wang at 14:47, 13 September 2012

2012-09-13T14:47:33Z

← Older revision		Revision as of 10:47, 13 September 2012
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	[[Image:12-240-Splash.png]]		[[12-240]][[Image:12-240-Splash.png]]

	[[12-240/Classnotes for Thursday September 13]]
	[[12-240/Classnotes for Tuesday September 11]]		[[12-240/Classnotes for Tuesday September 11]]

	In the first class, Professor says something about particular fields. Forgive me, because I am an international student, if I can not express information precisely.		In the first class, Professor says something about particular fields. Forgive me, because I am an international student, if I can not express information precisely.

			About: F(n)
⚫	~~About: F(n)~~ F(1) F(2) and F(3) are a field, but F(4) is not. Professor said that any number N which is not a prime number can not "form" a field.

		⚫	F(1) F(2) and F(3) are a field, but F(4) is not. Professor said that any number n which is not a prime number can not "form" a field F(n)

	If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"		If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"
Line 13:		Line 15:
	Here is the proof.		Here is the proof.

	If we have a field F(n), and n=ab (a,b,n∈ N, a,b≠1)		If we have a field F(n), and n=ab (a,b,n∈ N, a,b≠1) , which means n is not a prime number.

	IN defination of multiplication		IN defination of multiplication
Line 19:		Line 21:
	* 0 1 2 3 .......... b.......n-1		* 0 1 2 3 .......... b.......n-1

	0 . . . . ......................		0 0 0 0 0 ...........0.........0

	1 . . . . ......................		1 0 . . . ......................

	2 . . . . ......................		2 0 . . . ......................

	3 . . . . .......................		3 0 . . . .......................

	4 . . . . .......................		4 0 . . . .......................

	. . . . . .......................		.. 0 . . . .......................

	a 0 (a) (2a) (3a)...........(ab).....(n-1)a ( in this row, every element mod n)		a 0 (a) (2a) (3a)...........(ab).....(n-1)a ''' ( in this row, every element mod n)'''

	....................................		....................................
Line 41:		Line 43:
	.....................................		.....................................

	n-1..................................		(n-1) 0................................


	see the (a+1)th row		see the (a+1)th row

	There must be a "1" in this row, actually each row or column, to meet the rule :Existence of Negatives/Inverses.		There must be a "1" in this row, actually each row or column, to meet the rule :Existence of Negatives/Inverses. (About the rule, seen in the file "12-240/Classnotes for Tuesday September 11")



	So if F(n) is a field, then		So if F(n) is a field, then
	~~1.ma=kn+1 (k,m∈N*, m<n)~~ there must exist k,m.		there must exist k,m ∈N*, m<n
			to meet the equation:
	2.n=ab
			ma=kn+1


			And we know that n=ab


	~~==>>~~ma=ka*b+1 (a≠1)		So ma=ka*b+1 (a≠1)

	~~==>>~~m=k*b+1/a		Hence m=k*b+1/a

	unless a=1		unless a=1

Michael.Wang at 18:51, 12 September 2012

2012-09-12T18:51:46Z

← Older revision		Revision as of 14:51, 12 September 2012
Line 9:		Line 9:
	If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"		If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"

			'''Why all the numbers which are not prime numbers can not form a field F(n)?'''

	Here is the proof.		Here is the proof.

Michael.Wang at 18:48, 12 September 2012

2012-09-12T18:48:30Z

← Older revision		Revision as of 14:48, 12 September 2012
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			[[Image:12-240-Splash.png]]
			[[12-240/Classnotes for Thursday September 13]]
			[[12-240/Classnotes for Tuesday September 11]]

	In the first class, Professor says something about particular fields. Forgive me, because I am an international student, if I can not express information precisely.		In the first class, Professor says something about particular fields. Forgive me, because I am an international student, if I can not express information precisely.

	About: F(n) F(1) F(2) and F(3) are a field, but F(4) is not. Professor said that any number N which is not a prime number can not "form" a field. ~~We should ask why...~~		About: F(n) F(1) F(2) and F(3) are a field, but F(4) is not. Professor said that any number N which is not a prime number can not "form" a field.

			If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"

	Here is the proof.		Here is the proof.
Line 21:		Line 27:
	4 . . . . .......................		4 . . . . .......................

			. . . . . .......................
	.

	a 0 (a) (2a) (3a)...........(ab).....(n-1)a ( in this row, every element mod n)		a 0 (a) (2a) (3a)...........(ab).....(n-1)a ( in this row, every element mod n)

			....................................
	.

			....................................
	.

			......................................
	.

			.....................................
	.

	n-1..................................		n-1..................................
Line 38:		Line 44:
	see the (a+1)th row		see the (a+1)th row

	There must be a "1" in this row, each row ~~precisely~~ to meet the rule~~. ( The rule... you know, I cannot~~ ~~find~~ ~~some~~ ~~notations~~.)		There must be a "1" in this row, actually each row or column, to meet the rule :Existence of Negatives/Inverses.


⚫	If F(n) is a field, then

⚫	1.ma=kn+1 (k,m∈N*, m<n)
		⚫	So if F(n) is a field, then
		⚫	1.ma=kn+1 (k,m∈N*, m<n) there must exist k,m.
	2.n=ab		2.n=ab


⚫	==>>m=kb+1/a
			==>>ma=ka*b+1 (a≠1)

		⚫	==>>m=k*b+1/a

	unless a=1		unless a=1
	m will not exist.		m will not exist, because m should be an integer.
	so F(n), when n is not a prime number, is not a field.		so F(n), when n is not a prime number, is not a field.

Michael.Wang: 12-240/ moved to 12-240/Fields' Further proof

2012-09-12T15:55:17Z

12-240/ moved to 12-240/Fields' Further proof

← Older revision

Revision as of 11:55, 12 September 2012

Michael.Wang at 14:51, 12 September 2012

2012-09-12T14:51:44Z

← Older revision		Revision as of 10:51, 12 September 2012
Line 37:		Line 37:

	see the (a+1)th row		see the (a+1)th row

			There must be a "1" in this row, each row precisely to meet the rule. ( The rule... you know, I cannot find some notations.)

	If F(n) is a field, then		If F(n) is a field, then

Michael.Wang at 01:10, 12 September 2012

2012-09-12T01:10:11Z

New page

In the first class, Professor says something about particular fields. Forgive me, because I am an international student, if I can not express information precisely.

About: F(n) F(1) F(2) and F(3) are a field, but F(4) is not. Professor said that any number N which is not a prime number can not "form" a field. We should ask why...

Here is the proof.

If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1)

IN defination of multiplication

* 0 1 2 3 .......... b.......n-1

0 . . . . ......................

1 . . . . ......................

2 . . . . ......................

3 . . . . .......................

4 . . . . .......................

.

a 0 (a) (2a) (3a)...........(a*b).....(n-1)*a ( in this row, every element mod n)

.

.

.

.

n-1..................................

see the (a+1)th row

If F(n) is a field, then
1.m*a=k*n+1 (k,m∈N*, m<n)
2.n=ab

==>>m=kb+1/a

unless a=1
m will not exist.
so F(n), when n is not a prime number, is not a field.

There is a large need for me to improve my format. Editing is welcomed.

PS: But till now, there are still some questions existing.
How can we prove that a prime number N can absolutely form a field? Is there any exception?
I am still working on it.
----Michael.Wang