Talk:Near-field (mathematics)

Rewrite of article

The article was little more than a stub, and gave no references. It also contained at least one error, and some of the information it gave was arguably not in the most helpful form. I have rewritten it from scratch to give a fuller account.JamesBWatson (talk) 17:24, 1 December 2008 (UTC)

Equivalence of axioms

Can you elaborate on why "A2, A4, A5" can't be replaced with "A4*" in the definition? It seems clear to me that this can be done. Arun (talk) 22:32, 26 August 2009 (UTC)

If anyone else is interested in this, it occurs to me now that a much simpler answer than the one I originally gave (below) is this: Because A4* tells us nothing about multiplication by zero, whereas both A2 and A4 do so. JamesBWatson (talk) 10:58, 21 January 2010 (UTC)

Unless I have missed something, this is already answered in the article. The "exceptional structure" defined in the last sentence of "Notes on the definition" number 2 satisfies A1, A3, and A4* but clearly not A4 (consider $a=0$).

As for A2 (associativity of multiplication), it is independent of all the other axioms, as can be seen from the following example, which obeys all the axioms except A2. Take a field of order 4. Let i be an element of the field which is not equal to 0 or 1. Change the definition of multiplication by setting both i.0 and (1+i).0 equal to 1 instead of 0.

JamesBWatson (talk) 19:34, 31 August 2009 (UTC)