Problem

Source: Iranian National Olympiad (3rd Round) 2007

Tags: algebra, polynomial, quadratics, complex numbers, algebra proposed



Scientist have succeeded to find new numbers between real numbers with strong microscopes. Now real numbers are extended in a new larger system we have an order on it (which if induces normal order on $ \mathbb R$), and also 4 operations addition, multiplication,... and these operation have all properties the same as $ \mathbb R$. a) Prove that in this larger system there is a number which is smaller than each positive integer and is larger than zero. b) Prove that none of these numbers are root of a polynomial in $ \mathbb R[x]$.