부산대학교 도서관

For positive integers n ≥ m , let P n , m (x) : = ∑ j = 0 m n j x j = n 0 + n 1 x + ... + n m x m be the truncated binomial expansion of (1 + x) n consisting of all terms of degree ≤ m. It is conjectured that for n > m + 1 , the polynomial P n , m (x) is irreducible. We confirm this conjecture when 2 m ≤ n < (m + 1) 1 0. Also we show for any r ≥ 1 0 and 2 m ≤ n < (m + 1) r + 1 , the polynomial P n , m (x) is irreducible when m ≥ max { 1 0 6 , 2 r 3 }. Under the explicit abc-conjecture, for a fixed m , we give an explicit n 0 , n 1 depending only on m such that ∀ n ≥ n 0 , the polynomial P n , m (x) is irreducible. Further ∀ n ≥ n 1 , the Galois group associated to P n , m (x) is the symmetric group S m. [ABSTRACT FROM AUTHOR]