Let A be an integral domain of characteristic 5. Prove that x+4 is a factor of (x^m)+4 in A[X] for every m
Since A has characteristic 5, 5a = 0 for all a in A. Let m be a positive integer. Then in A[x],
x^m + 4 = x^m - 1 = (x - 1)(x^(m - 1) + ••• + x + 1) = (x + 4)(x^(m-1) + ••• + x + 1),
showing that x + 4 is a factor of x^m + 4.
Comments
Since A has characteristic 5, 5a = 0 for all a in A. Let m be a positive integer. Then in A[x],
x^m + 4 = x^m - 1 = (x - 1)(x^(m - 1) + ••• + x + 1) = (x + 4)(x^(m-1) + ••• + x + 1),
showing that x + 4 is a factor of x^m + 4.