http://drorbn.net/api.php?action=feedcontributions&user=Michael.Wang&feedformat=atomDrorbn - User contributions [en]2024-03-28T08:40:22ZUser contributionsMediaWiki 1.21.1http://drorbn.net/index.php?title=12-240/Class_Photo12-240/Class Photo2012-09-29T19:52:20Z<p>Michael.Wang: /* Who We Are... */</p>
<hr />
<div>Our class on September 25, 2012:<br />
<br />
[[Image:12-240-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]<br />
{{12-240/Navigation}}<br />
<br />
Please identify yourself in this photo! There are two ways to do that:<br />
<br />
* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.<br />
* Send [[User:Drorbn|Dror]] an email message with this information.<br />
<br />
The first option is more fun but less private.<br />
<br />
===Who We Are...===<br />
<br />
{| align=center border=1 cellspacing=0<br />
|-<br />
!First name<br />
!Last name<br />
!User ID<br />
!Email<br />
!Place in photo<br />
!Comments<br />
<br />
{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math. toronto. edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}<br />
{{Photo Entry|last=Frailich|first=Rebecca|userid=Rebecca.frailich|email=rebecca.frailich@ mail.utoronto.ca|location=Last row, in between two guys standing at the back (one in red, one in black) |comments=}}<br />
{{Photo Entry|last=Hoover|first=Ken|userid=Khoover|email=ken.hoover@ mail.utoronto.ca|location=First row, fourth from the right.|comments=}}<br />
{{Photo Entry|last=Klingspor|first=Josefine|userid=Josefine|email=josefine. klingspor@ mail. utoronto. ca|location=First row, second from left.|comments=}}<br />
{{Photo Entry|last=Le|first=Quan|userid=Quanle|email=quan.le@mail.utoronto.ca|location=Start bottom right corner, third from right. Go three steps north-west. Directly north-east from there, in blue collar shirt|comments=}}<br />
{{Photo Entry|last=Liu|first=Zhaowei|userid=tod|email=tod. liu@ mail. utoronto .ca|location=First row, third from the right|comments=}}<br />
{{Photo Entry|last=Millson|first=Richard|userid=Richardm|email=r.millson@ mail. utoronto. ca|location=Seventh row from the front, fourth from the right, blue sweater|comments=}}<br />
{{Photo Entry|last=McGrath|first=Celton|userid=CeltonMcGrath|email=celton. mcgrath@ mail. utoronto. ca|location=4th row front from, centre right, brown sweater|comments=}}<br />
{{Photo Entry|last=Morenz|first=Karen|userid=KJMorenz|email=kjmorenz@ gmail.com|location=3rd-ish row from the back, centre right, purple shirt|comments=}}<br />
{{Photo Entry|last=Vicencio-Heap|first=Felipe|userid=Heapfeli|email=felipe. vicencio. heap@ mail. utoronto. ca|location=Second row from the front, furthest to the right.|comments=}}<br />
{{Photo Entry|last=Wamer|first=Kyle|userid=kylewamer|email=kyle. wamer @ mail. utoronto. ca|location=Second row, fifth from the left in the red shirt.|comments=}}<br />
{{Photo Entry|last=Yang|first=Chen|userid=chen|email=neochen. yang@ mail. utoronto. ca|location=sixth row, first from the right in the black pull-over.|comments=}}<br />
{{Photo Entry|last=Zhang|first=BingZhen|userid=Zetalda|email=bingzhen.zhang@ mail. utoronto. ca|location=Second last row, third from left.|comments=}}<br />
{{Photo Entry|last=Zhao|first=TianChen|userid=Ericolony|email=zhao_tianchen@ hotmail. com|location=fourth row, the guy in green shirt.|comments=}}<br />
{{Photo Entry|last=Zibert|first=Vincent|userid=vincezibert|email=vincent. zibert@ mail. utoronto. ca|location=Directly beneath the white notice posted on the door on the right-hand side.|comments=}}<br />
{{Photo Entry|last=Zoghi|first=Sina|userid=sina.zoghi|email=sina.zoghi@ utoronto .ca|location=First row, leftest left.|comments=}}<br />
{{Photo Entry|last=Léger|first=Zacharie|userid=zach.leger8|email=zacharie. leger@ mail. utronto. ca|location= 5th row in a black T-shirt.|comments=}}<br />
{{Photo Entry|last=Wang|first=Minqi|userid=Michael.Wang|email=wangminqi@ yahoo.cn|location=First row, fourth from the left in black oufit) |comments=}}<br />
<br />
<br />
<!--PLEASE KEEP IN ALPHABETICAL ORDER, BY LAST NAME--><br />
|}<br />
<br />
<!--PLEASE KEEP IN ALPHABETICAL ORDER, BY LAST NAME--></div>Michael.Wanghttp://drorbn.net/index.php?title=12-240/Fields%27_Further_proof12-240/Fields' Further proof2012-09-16T20:13:15Z<p>Michael.Wang: </p>
<hr />
<div>[[12-240]][[Image:12-240-Splash.png]]<br />
<br />
[[12-240/Classnotes for Tuesday September 11]]<br />
<br />
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.<br />
<br />
About: F(n) <br />
<br />
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)<br />
<br />
If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"<br />
<br />
'''Why all the numbers which are not prime numbers can not form a field F(n)?'''<br />
<br />
Here is the proof. <br />
<br />
If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1) , which means n is not a prime number.<br />
<br />
IN defination of multiplication <br />
<br />
* 0 1 2 3 .......... b.......n-1 <br />
<br />
0 0 0 0 0 ...........0........0 <br />
<br />
1 0 . . . ......................<br />
<br />
2 0 . . . ......................<br />
<br />
3 0 . . . ...................... <br />
<br />
4 0 . . . ......................<br />
<br />
.. 0 . . . ......................<br />
<br />
a 0 (a) (2a) (3a).........(a*b)....(n-1)*a ''' ( in this row, every element mod n)''' <br />
<br />
....0.....................................<br />
<br />
....0.....................................<br />
<br />
....0.....................................<br />
<br />
....0.....................................<br />
<br />
(n-1) 0...................................<br />
see the (a+1)th row <br />
<br />
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")<br />
<br />
<br />
<br />
So if F(n) is a field, then <br />
there must exist k,m ∈N*, m<n<br />
to meet the equation:<br />
m*a=k*n+1<br />
<br />
<br />
And we know that n=ab<br />
<br />
<br />
So m*a=k*a*b+1 (a≠1)<br />
<br />
Hence m=k*b+1/a <br />
<br />
unless a=1 <br />
m will not exist, because m should be an integer.<br />
so F(n), when n is not a prime number, is not a field.<br />
<br />
There is a large need for me to improve my format. Editing is welcomed.<br />
<br />
<br />
<br />
PS: But till now, there are still some questions existing.<br />
How can we prove that a prime number N can absolutely form a field? Is there any exception?<br />
I am still working on it.<br />
----Michael.Wang</div>Michael.Wanghttp://drorbn.net/index.php?title=12-240/Fields%27_Further_proof12-240/Fields' Further proof2012-09-16T20:11:06Z<p>Michael.Wang: </p>
<hr />
<div>[[12-240]][[Image:12-240-Splash.png]]<br />
<br />
[[12-240/Classnotes for Tuesday September 11]]<br />
<br />
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.<br />
<br />
About: F(n) <br />
<br />
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)<br />
<br />
If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"<br />
<br />
'''Why all the numbers which are not prime numbers can not form a field F(n)?'''<br />
<br />
Here is the proof. <br />
<br />
If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1) , which means n is not a prime number.<br />
<br />
IN defination of multiplication <br />
<br />
* 0 1 2 3 .......... b.......n-1 <br />
<br />
0 0 0 0 0 ...........0........0 <br />
<br />
1 0 . . . ......................<br />
<br />
2 0 . . . ......................<br />
<br />
3 0 . . . ...................... <br />
<br />
4 0 . . . ......................<br />
<br />
. 0 . . . ......................<br />
<br />
a 0 (a) (2a) (3a).........(a*b)....(n-1)*a ''' ( in this row, every element mod n)''' <br />
<br />
....0.....................................<br />
<br />
....0.....................................<br />
<br />
....0.....................................<br />
<br />
....0.....................................<br />
<br />
(n-1) 0...................................<br />
see the (a+1)th row <br />
<br />
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")<br />
<br />
<br />
<br />
So if F(n) is a field, then <br />
there must exist k,m ∈N*, m<n<br />
to meet the equation:<br />
m*a=k*n+1<br />
<br />
<br />
And we know that n=ab<br />
<br />
<br />
So m*a=k*a*b+1 (a≠1)<br />
<br />
Hence m=k*b+1/a <br />
<br />
unless a=1 <br />
m will not exist, because m should be an integer.<br />
so F(n), when n is not a prime number, is not a field.<br />
<br />
There is a large need for me to improve my format. Editing is welcomed.<br />
<br />
<br />
<br />
PS: But till now, there are still some questions existing.<br />
How can we prove that a prime number N can absolutely form a field? Is there any exception?<br />
I am still working on it.<br />
----Michael.Wang</div>Michael.Wanghttp://drorbn.net/index.php?title=12-240/Fields%27_Further_proof12-240/Fields' Further proof2012-09-16T20:09:20Z<p>Michael.Wang: </p>
<hr />
<div><nowiki>[[12-240]][[Image:12-240-Splash.png]]<br />
<br />
[[12-240/Classnotes for Tuesday September 11]]<br />
<br />
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.<br />
<br />
About: F(n) <br />
<br />
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)<br />
<br />
If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"<br />
<br />
'''Why all the numbers which are not prime numbers can not form a field F(n)?'''<br />
<br />
Here is the proof. <br />
<br />
If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1) , which means n is not a prime number.<br />
<br />
IN defination of multiplication <br />
<br />
* 0 1 2 3 .......... b.......n-1 <br />
<br />
0 0 0 0 0 ...........0........0 <br />
<br />
1 0 . . . ......................<br />
<br />
2 0 . . . ......................<br />
<br />
3 0 . . . ...................... <br />
<br />
4 0 . . . ......................<br />
<br />
. 0 . . . ......................<br />
<br />
a 0 (a) (2a) (3a).........(a*b)....(n-1)*a ''' ( in this row, every element mod n)''' <br />
<br />
....0.....................................<br />
<br />
....0.....................................<br />
<br />
....0.....................................<br />
<br />
....0.....................................<br />
<br />
(n-1) 0...................................<br />
see the (a+1)th row <br />
<br />
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")<br />
<br />
<br />
<br />
So if F(n) is a field, then <br />
there must exist k,m ∈N*, m<n<br />
to meet the equation:<br />
m*a=k*n+1<br />
<br />
<br />
And we know that n=ab<br />
<br />
<br />
So m*a=k*a*b+1 (a≠1)<br />
<br />
Hence m=k*b+1/a <br />
<br />
unless a=1 <br />
m will not exist, because m should be an integer.<br />
so F(n), when n is not a prime number, is not a field.<br />
<br />
There is a large need for me to improve my format. Editing is welcomed.<br />
<br />
<br />
<br />
PS: But till now, there are still some questions existing.<br />
How can we prove that a prime number N can absolutely form a field? Is there any exception?<br />
I am still working on it.<br />
----Michael.Wang<br />
<nowiki></nowiki></div>Michael.Wanghttp://drorbn.net/index.php?title=12-240/Fields%27_Further_proof12-240/Fields' Further proof2012-09-13T14:47:33Z<p>Michael.Wang: </p>
<hr />
<div>[[12-240]][[Image:12-240-Splash.png]]<br />
<br />
[[12-240/Classnotes for Tuesday September 11]]<br />
<br />
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.<br />
<br />
About: F(n) <br />
<br />
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)<br />
<br />
If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"<br />
<br />
'''Why all the numbers which are not prime numbers can not form a field F(n)?'''<br />
<br />
Here is the proof. <br />
<br />
If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1) , which means n is not a prime number.<br />
<br />
IN defination of multiplication <br />
<br />
* 0 1 2 3 .......... b.......n-1 <br />
<br />
0 0 0 0 0 ...........0.........0 <br />
<br />
1 0 . . . ...................... <br />
<br />
2 0 . . . ...................... <br />
<br />
3 0 . . . ....................... <br />
<br />
4 0 . . . ....................... <br />
<br />
.. 0 . . . .......................<br />
<br />
a 0 (a) (2a) (3a)...........(a*b).....(n-1)*a ''' ( in this row, every element mod n)''' <br />
<br />
....................................<br />
<br />
....................................<br />
<br />
......................................<br />
<br />
.....................................<br />
<br />
(n-1) 0................................ <br />
<br />
<br />
see the (a+1)th row <br />
<br />
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")<br />
<br />
<br />
<br />
So if F(n) is a field, then <br />
there must exist k,m ∈N*, m<n<br />
to meet the equation:<br />
m*a=k*n+1<br />
<br />
<br />
And we know that n=ab<br />
<br />
<br />
So m*a=k*a*b+1 (a≠1)<br />
<br />
Hence m=k*b+1/a <br />
<br />
unless a=1 <br />
m will not exist, because m should be an integer.<br />
so F(n), when n is not a prime number, is not a field.<br />
<br />
There is a large need for me to improve my format. Editing is welcomed.<br />
<br />
<br />
<br />
PS: But till now, there are still some questions existing.<br />
How can we prove that a prime number N can absolutely form a field? Is there any exception?<br />
I am still working on it.<br />
----Michael.Wang</div>Michael.Wanghttp://drorbn.net/index.php?title=12-240/Fields%27_Further_proof12-240/Fields' Further proof2012-09-12T18:51:46Z<p>Michael.Wang: </p>
<hr />
<div>[[Image:12-240-Splash.png]]<br />
[[12-240/Classnotes for Thursday September 13]]<br />
[[12-240/Classnotes for Tuesday September 11]]<br />
<br />
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.<br />
<br />
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. <br />
<br />
If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"<br />
<br />
'''Why all the numbers which are not prime numbers can not form a field F(n)?'''<br />
<br />
Here is the proof. <br />
<br />
If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1) <br />
<br />
IN defination of multiplication <br />
<br />
* 0 1 2 3 .......... b.......n-1 <br />
<br />
0 . . . . ...................... <br />
<br />
1 . . . . ...................... <br />
<br />
2 . . . . ...................... <br />
<br />
3 . . . . ....................... <br />
<br />
4 . . . . ....................... <br />
<br />
. . . . . .......................<br />
<br />
a 0 (a) (2a) (3a)...........(a*b).....(n-1)*a ( in this row, every element mod n) <br />
<br />
....................................<br />
<br />
....................................<br />
<br />
......................................<br />
<br />
.....................................<br />
<br />
n-1.................................. <br />
<br />
<br />
see the (a+1)th row <br />
<br />
There must be a "1" in this row, actually each row or column, to meet the rule :Existence of Negatives/Inverses. <br />
<br />
<br />
<br />
So if F(n) is a field, then <br />
1.m*a=k*n+1 (k,m∈N*, m<n) there must exist k,m.<br />
2.n=ab<br />
<br />
<br />
==>>m*a=k*a*b+1 (a≠1)<br />
<br />
==>>m=k*b+1/a <br />
<br />
unless a=1 <br />
m will not exist, because m should be an integer.<br />
so F(n), when n is not a prime number, is not a field.<br />
<br />
There is a large need for me to improve my format. Editing is welcomed.<br />
<br />
<br />
<br />
PS: But till now, there are still some questions existing.<br />
How can we prove that a prime number N can absolutely form a field? Is there any exception?<br />
I am still working on it.<br />
----Michael.Wang</div>Michael.Wanghttp://drorbn.net/index.php?title=12-240/Fields%27_Further_proof12-240/Fields' Further proof2012-09-12T18:48:30Z<p>Michael.Wang: </p>
<hr />
<div>[[Image:12-240-Splash.png]]<br />
[[12-240/Classnotes for Thursday September 13]]<br />
[[12-240/Classnotes for Tuesday September 11]]<br />
<br />
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.<br />
<br />
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. <br />
<br />
If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"<br />
<br />
Here is the proof. <br />
<br />
If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1) <br />
<br />
IN defination of multiplication <br />
<br />
* 0 1 2 3 .......... b.......n-1 <br />
<br />
0 . . . . ...................... <br />
<br />
1 . . . . ...................... <br />
<br />
2 . . . . ...................... <br />
<br />
3 . . . . ....................... <br />
<br />
4 . . . . ....................... <br />
<br />
. . . . . .......................<br />
<br />
a 0 (a) (2a) (3a)...........(a*b).....(n-1)*a ( in this row, every element mod n) <br />
<br />
....................................<br />
<br />
....................................<br />
<br />
......................................<br />
<br />
.....................................<br />
<br />
n-1.................................. <br />
<br />
<br />
see the (a+1)th row <br />
<br />
There must be a "1" in this row, actually each row or column, to meet the rule :Existence of Negatives/Inverses. <br />
<br />
<br />
<br />
So if F(n) is a field, then <br />
1.m*a=k*n+1 (k,m∈N*, m<n) there must exist k,m.<br />
2.n=ab<br />
<br />
<br />
==>>m*a=k*a*b+1 (a≠1)<br />
<br />
==>>m=k*b+1/a <br />
<br />
unless a=1 <br />
m will not exist, because m should be an integer.<br />
so F(n), when n is not a prime number, is not a field.<br />
<br />
There is a large need for me to improve my format. Editing is welcomed.<br />
<br />
<br />
<br />
PS: But till now, there are still some questions existing.<br />
How can we prove that a prime number N can absolutely form a field? Is there any exception?<br />
I am still working on it.<br />
----Michael.Wang</div>Michael.Wanghttp://drorbn.net/index.php?title=12-240/Fields%27_Further_proof12-240/Fields' Further proof2012-09-12T15:55:17Z<p>Michael.Wang: 12-240/ moved to 12-240/Fields' Further proof</p>
<hr />
<div>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.<br />
<br />
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...<br />
<br />
Here is the proof. <br />
<br />
If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1) <br />
<br />
IN defination of multiplication <br />
<br />
* 0 1 2 3 .......... b.......n-1 <br />
<br />
0 . . . . ...................... <br />
<br />
1 . . . . ...................... <br />
<br />
2 . . . . ...................... <br />
<br />
3 . . . . ....................... <br />
<br />
4 . . . . ....................... <br />
<br />
.<br />
<br />
a 0 (a) (2a) (3a)...........(a*b).....(n-1)*a ( in this row, every element mod n) <br />
<br />
.<br />
<br />
.<br />
<br />
.<br />
<br />
.<br />
<br />
n-1.................................. <br />
<br />
<br />
see the (a+1)th row <br />
<br />
There must be a "1" in this row, each row precisely to meet the rule. ( The rule... you know, I cannot find some notations.) <br />
<br />
If F(n) is a field, then <br />
1.m*a=k*n+1 (k,m∈N*, m<n)<br />
2.n=ab<br />
<br />
==>>m=kb+1/a <br />
<br />
unless a=1 <br />
m will not exist.<br />
so F(n), when n is not a prime number, is not a field.<br />
<br />
There is a large need for me to improve my format. Editing is welcomed.<br />
<br />
<br />
<br />
PS: But till now, there are still some questions existing.<br />
How can we prove that a prime number N can absolutely form a field? Is there any exception?<br />
I am still working on it.<br />
----Michael.Wang</div>Michael.Wanghttp://drorbn.net/index.php?title=12-240/12-240/2012-09-12T15:55:17Z<p>Michael.Wang: 12-240/ moved to 12-240/Fields' Further proof: Lack title</p>
<hr />
<div>#redirect [[12-240/Fields' Further proof]]</div>Michael.Wanghttp://drorbn.net/index.php?title=12-240/Fields%27_Further_proof12-240/Fields' Further proof2012-09-12T14:51:44Z<p>Michael.Wang: </p>
<hr />
<div>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.<br />
<br />
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...<br />
<br />
Here is the proof. <br />
<br />
If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1) <br />
<br />
IN defination of multiplication <br />
<br />
* 0 1 2 3 .......... b.......n-1 <br />
<br />
0 . . . . ...................... <br />
<br />
1 . . . . ...................... <br />
<br />
2 . . . . ...................... <br />
<br />
3 . . . . ....................... <br />
<br />
4 . . . . ....................... <br />
<br />
.<br />
<br />
a 0 (a) (2a) (3a)...........(a*b).....(n-1)*a ( in this row, every element mod n) <br />
<br />
.<br />
<br />
.<br />
<br />
.<br />
<br />
.<br />
<br />
n-1.................................. <br />
<br />
<br />
see the (a+1)th row <br />
<br />
There must be a "1" in this row, each row precisely to meet the rule. ( The rule... you know, I cannot find some notations.) <br />
<br />
If F(n) is a field, then <br />
1.m*a=k*n+1 (k,m∈N*, m<n)<br />
2.n=ab<br />
<br />
==>>m=kb+1/a <br />
<br />
unless a=1 <br />
m will not exist.<br />
so F(n), when n is not a prime number, is not a field.<br />
<br />
There is a large need for me to improve my format. Editing is welcomed.<br />
<br />
<br />
<br />
PS: But till now, there are still some questions existing.<br />
How can we prove that a prime number N can absolutely form a field? Is there any exception?<br />
I am still working on it.<br />
----Michael.Wang</div>Michael.Wanghttp://drorbn.net/index.php?title=12-240/Fields%27_Further_proof12-240/Fields' Further proof2012-09-12T01:10:11Z<p>Michael.Wang: </p>
<hr />
<div>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.<br />
<br />
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...<br />
<br />
Here is the proof. <br />
<br />
If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1) <br />
<br />
IN defination of multiplication <br />
<br />
* 0 1 2 3 .......... b.......n-1 <br />
<br />
0 . . . . ...................... <br />
<br />
1 . . . . ...................... <br />
<br />
2 . . . . ...................... <br />
<br />
3 . . . . ....................... <br />
<br />
4 . . . . ....................... <br />
<br />
.<br />
<br />
a 0 (a) (2a) (3a)...........(a*b).....(n-1)*a ( in this row, every element mod n) <br />
<br />
.<br />
<br />
.<br />
<br />
.<br />
<br />
.<br />
<br />
n-1.................................. <br />
<br />
<br />
see the (a+1)th row <br />
<br />
If F(n) is a field, then <br />
1.m*a=k*n+1 (k,m∈N*, m<n)<br />
2.n=ab<br />
<br />
==>>m=kb+1/a <br />
<br />
unless a=1 <br />
m will not exist.<br />
so F(n), when n is not a prime number, is not a field.<br />
<br />
There is a large need for me to improve my format. Editing is welcomed.<br />
<br />
<br />
<br />
PS: But till now, there are still some questions existing.<br />
How can we prove that a prime number N can absolutely form a field? Is there any exception?<br />
I am still working on it.<br />
----Michael.Wang</div>Michael.Wang