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0(x log 0(x log x 2x log absolute value algebraic integers algebraic number field analogue belongs bounded choose coefficients of g(x complex numbers conjugates of 9 consider contains Corollary cubes define Definition discriminant divide element elementary number theory equal equation equivalent estimate exists an algebraic expressed field K(9 finite number fixed algebraic number fractional ideal function greatest common divisor Hence ideal class identity implies inequality integral ideal interval inversion formula lattice points leading coefficient linear combination linearly independent log2 log2N log2x minimal basis multiplication n<x n non-zero number of ideals obtain polynomial f(x prime number theorem prime theorem principal ideal prove rational coefficients rational numbers real numbers residue classes modulo result right side root of unity satisfied by 9 Similarly summation suppose Theorem 1.4 unique factorization theorem vectors write written