Write down a polynomial of degree 4 with integer coefficients, so that two of the roots are 1 - sqrt(2) and 3 + sqrt(3).
+128475 The conjugates of these are also roots
( x - (1 - sqrt (2)) ( x - (1 + sqrt (2) ) =
x^2 -x( 1-sqrt (2)) - x ( 1 + sqrt (2) + ( 1-sqrt (2))(1 +sqrt (2))
x^2 -2 x + 1 -2 =
(x^2 -2x -1)
And
(x - (3 + sqrt (3)) ( x - (3 + sqrt (3)) =
x^2 -x ( 3 + sqrt (3)) - x (3 -sqrt(3)) + (3 -sqrt (3)) (3 + sqrt (3))
x^2 - 6x + 9 - 3 =
(x^2 - 6x + 6)
So
(x^2 - 2x - 1) (x^2 - 6x + 6) =
x^4 - 6x^3 + 6x^2
-2x^3 + 12x^2 - 12x
-x^2 +6x - 6
_________________________
x^4 - 8x^3 + 17x^2 - 6x - 6
