The Square Threshold Problem in Number Fields

Lafferty, Matt

MS

Mathematical Science

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Kevin L James

Let K be a degree n extension of Q, and let O_K be the ring of algebraic integers in K. Let x >= 2. Suppose we were to generate an ideal sequence by choosing ideals with norm at most x from O_K, independently and with uniform probability. How long would our sequence of ideals need to be before we obtain a subsequence whose terms have a product that is a square ideal in O_K? We show that the answer is about exp((2ln(x)lnln(x))^(1/2)).