Question 28
Let $S = \{0, 95\}$. In each step, Lucija is extending the set $S$ in the following way. She chooses a polynomial with coefficients in $S$, distinct from zero, and extends $S$ with all integer roots of a chosen polynomial. She repeats the procedure by choosing another polynomial with coefficients from the extended set $S$ as long as she can get new roots.
Prove that Lucija can, in a finite number of steps, extend the set $S$ up to the set which is no longer extensible. How many elements does the set $S$ have at the end?
Prove that Lucija can, in a finite number of steps, extend the set $S$ up to the set which is no longer extensible. How many elements does the set $S$ have at the end?