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We show

forallxinQ:x*x!=2.

From the construction of **Q**, we
know `x` = a/b for some `a` in **Z** and `b`
in **Z**_{> 0}
such that (2) N(`a`) and
N(`b`) are relatively prime.
We have `a` *_{Z}
`a`/`b` *_{Z} `b` = 2 and thus
(from now on we operate in **Z** and drop the corresponding subscripts):

(3)From (3) we know N(2) | N(a*a= 2 *b*b.

(5)From (3) and (5) we have 2*a= 2*c.

Author: Wolfgang Schreiner

Last Modification: November 30, 1999