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Talk:Elementary proof (view source)
Revision as of 01:44, February 7, 2007
, 01:44, February 7, 2007uniqueness
--[[User:Aschlafly|Aschlafly]] 20:12, 6 February 2007 (EST)
--[[User:Aschlafly|Aschlafly]] 20:12, 6 February 2007 (EST)
: Existence depends on what system you are operating in. Uniqueness is fairly easy if you are operating in a field. Proposition: in any field F, with c and element of F, the equation <math>x^2=c</math> has at most 2 solutions. Proof: Consider <math>x^2=c</math>. this implies that <math>x^2-c=0</math>. We may assume that c has at least one square root (call it <math>c^{1/2}</math>). So we have <math>(x+c^{1/2})(x-c^{1/2})</math>. Now, since fields have no zero divisors (easy excercise), we must have <math>x+c^{1/2}=0</math> or <math>x-c^{1/2}=0</math> which gives us only two choices. Q.E.D. [[User:JoshuaZ|JoshuaZ]] 20:44, 6 February 2007 (EST)