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Printjmc
algebra intermediate
Problem
A set is constructed as follows. To begin, . Repeatedly, as long as possible, if is an integer root of some nonzero polynomial for some , all of whose coefficients are elements of , then is put into . When no more elements can be added to , how many elements does have?
Solution
We start with the set We can construct the polynomial which has as a root. Thus, we can expand our set to
We can then construct the polynomial which has as a root, and we can construct the polynomial which has as a root. Thus, we can expand our set to
Next, we can construct the polynomial which has as a root, the polynomial which has as a root, and the polynomial which has as a root. Our set is now
Finally, we can construct the polynomial which has as a root, giving us the set Now, suppose we construct the polynomial with coefficients from the set If then we can factor out some power of to obtain a polynomial where the constant term is nonzero. Thus, we can assume that
By the Integer Root Theorem, any integer root of this polynomial must divide But we see that any divisor of a non-zero element in already lies in so we cannot expand the set any further. Hence, the answer is elements.
We can then construct the polynomial which has as a root, and we can construct the polynomial which has as a root. Thus, we can expand our set to
Next, we can construct the polynomial which has as a root, the polynomial which has as a root, and the polynomial which has as a root. Our set is now
Finally, we can construct the polynomial which has as a root, giving us the set Now, suppose we construct the polynomial with coefficients from the set If then we can factor out some power of to obtain a polynomial where the constant term is nonzero. Thus, we can assume that
By the Integer Root Theorem, any integer root of this polynomial must divide But we see that any divisor of a non-zero element in already lies in so we cannot expand the set any further. Hence, the answer is elements.
Final answer
9