Tossup

Bounds on rational approximations of numbers with this property were refined by Thue, Siegel, and Roth. Bounds on approximations (10[1])of numbers with this property use that this property is [emphasize] not possessed by the infinite sum of one over “10 to the k (10[1])factorial.” If a-sub-i (10[2])are distinct (-5[1])numbers with this property, (10[1])then “e to the a-sub-i” (-5[1])are independent (-5[1])over the rationals. A field extension has this property if it has (10[1])finite degree. (10[1])A number (10[1])was shown [emphasize] not to have this property when a simplified version of the Lindemann–Weierstrass (-5[1])theorem was used (10[1])by Charles (10[1])Hermite. (10[1])All rationals and some irrationals like the golden (10[1])ratio belong (10[1])to the countable set of numbers with this (10[1])property. (-5[1])For 10 points, name this property of numbers that are the roots of polynomials (10[1]-5[1])with (-5[1])rational coefficients, which (10[1])is (10[1])contrasted with being (10[1])transcendental. ■END■ (10[5])

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