My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum


Thanks Tree5Thanks
Reply
 
LinkBack Thread Tools Display Modes
July 7th, 2018, 04:08 PM   #11
Math Team
 
Joined: Jan 2015
From: Alabama

Posts: 3,244
Thanks: 887

Quote:
Originally Posted by Maschke View Post
I did not say that at all. A field is required to have $0 \neq 1$. I merely pointed out that mathman's third line was unjustified.
If $0 \neq 1$, then 1 is NOT the additive identity so $x+ 1\ne x$ is indeed justified.
Country Boy is offline  
 
July 7th, 2018, 04:17 PM   #12
Senior Member
 
Joined: Aug 2012

Posts: 1,960
Thanks: 547

Quote:
Originally Posted by Country Boy View Post
If $0 \neq 1$, then 1 is NOT the additive identity so $x+ 1\ne x$ is indeed justified.
Right. $x \neq x + 1$ is a CONSEQUENCE of $0 \neq 1$.

But mathman used $x \neq x + 1$ to attempt to PROVE that $0 \neq 1$. He assumed the thing he was trying to prove.
Maschke is offline  
July 7th, 2018, 07:20 PM   #13
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: USA

Posts: 2,009
Thanks: 1042

Quote:
Originally Posted by Maschke View Post
I did not say that at all.
I didn't say you did.
romsek is offline  
July 7th, 2018, 08:53 PM   #14
Senior Member
 
Joined: Aug 2012

Posts: 1,960
Thanks: 547

Quote:
Originally Posted by romsek View Post
I didn't say you did.
Sorry. Trying to think about 0 = 1 makes my head hurt.

It turns out, by the way, that there's a thing called "the field with one element." Such a thing does not exist; but many abstract areas of math act as if it "should" exist. I don't really understand much about this but here it is.

https://en.wikipedia.org/wiki/Field_with_one_element

This is a pretty good article. It gives the flavor of this amazing out-there stuff whether or not you follow the details. Nobody can construct or even define a field with one element, but if they had one it would do all kinds of great things, even perhaps solve the Riemann hypothesis.

Last edited by Maschke; July 7th, 2018 at 09:00 PM.
Maschke is offline  
July 8th, 2018, 12:50 PM   #15
Global Moderator
 
Joined: May 2007

Posts: 6,540
Thanks: 591

Quote:
Originally Posted by Maschke View Post
How do you get line three, $x+1\ne x$? Aren't you assuming the thing you're trying to prove? If $0 = 1$ then $x + 1 = x$ and there is no contradiction.
I guess I overdid it. $x\times 1=x, x\times 0=0$. Both will be true only if $x=0$. I don't think much is done with a field consisting of one element .
mathman is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
field



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
does cyclic field implies Galois field rayman Abstract Algebra 1 March 11th, 2014 03:27 PM
Fin. gen field extension -> intermediate field f.g. also? watson Abstract Algebra 1 September 20th, 2012 06:53 PM
Show R (comm. domain) over a field k is a field if dimR<inft watson Abstract Algebra 1 September 14th, 2012 09:07 PM
Every quadratic field is contained in a cyclotomic field brunojo Abstract Algebra 0 June 5th, 2009 06:25 PM
field goodfeeling Algebra 0 December 31st, 1969 04:00 PM





Copyright © 2018 My Math Forum. All rights reserved.