
Abstract Algebra Abstract Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
October 19th, 2007, 04:45 PM  #1 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  ONAG, field question
So, I'm trying to force my way through On Numbers and Games with a very limited understanding of Algebra (I do this to myself way too often...) and I'm not understanding something. Conway is defining multiplication on On2, first by showing how it works with specific numbers. When he gets to 4.2, I get confused. He writes "As for 4.2, this cannot be 0,1,2, or 3, since we already know that these numbers form a subfield not containing 4. Similarly 4.2 cannot be one of 4,5 6, or 7 because that would make 4.3 one of 0,1,2, or 3." I understand what it means that 0,1,2,3 are a subfield, but I don't understand why they are a subfield. I also don't understand how 4.2 ∈ {4,5,6,7} > 4.3 ∈ {0,1,2,3} Thanks for the help. Or maybe I'm farther over my head than I thought? 
October 19th, 2007, 05:58 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
Not having that book, can you define the field for us?

October 20th, 2007, 11:15 AM  #3 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  
October 20th, 2007, 10:20 PM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
Had you used the term nimbers I would have understood. I have Conway's The Book of Numbers which describes both nimbers and surreal numbers. (I also have Knuth's original book on surreal numbers.) Let F = {0, 1, 2, 3} be a subfield of the nimbers and suppose 4 · 2 = f in F. Since f is in F, f · 2^(1) is in F. But f ·2^(1) = 4 · 2 · 2^(1) = 4, a contradiction. 
October 21st, 2007, 07:39 AM  #5 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 
Aha! Thank you, sir.

October 21st, 2007, 08:15 PM  #6 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Tips hat 

Tags 
field, onag, question 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Show R (comm. domain) over a field k is a field if dimR<inft  watson  Abstract Algebra  1  September 14th, 2012 10:07 PM 
Finite field question  watson  Abstract Algebra  5  January 11th, 2012 01:20 PM 
Elementary Field Question  gwsinger  Real Analysis  7  August 11th, 2011 02:43 PM 
Question about the field axioms  Touya Akira  Abstract Algebra  3  February 7th, 2011 05:50 AM 
Sigma Field Question  NightBlues  Advanced Statistics  0  September 6th, 2010 06:54 PM 