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October 19th, 2007, 03:45 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, 04:58 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, 10:15 AM  #3 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  
October 20th, 2007, 09:20 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, 06:39 AM  #5 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 
Aha! Thank you, sir.

October 21st, 2007, 07:15 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 

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