1. The problem statement, all variables and given/known data Prove that the kernel of the homomorphism Z[x]->R sending x to 1+sqrt(2) is a principle ideal, and find a generator for this ideal. Z is the integers R is the real numbers 3. The attempt at a solution I assume sending x to sqrt(2) is an example. We should first find the actual map shoudn't we? Otherwise we can't find the kernel of this map. I have tried like 1. Sending the coefficients of x to C+sqrt(x) or C(1+sqrt(2)) where C is the coefficient of x. 2. Sending the constant of the polynomial to C+1+sqrt(2) or C(1+sqrt(2)) where C is the consant of the polynomial. 3. Sending the coefficient of the greatest power in the polynomial to C+sqrt(2) or C(1+sqrt(2)) where C is the coefficient of the greatest power of the polynomial. All 3 example maps do not form a homomorphism.