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
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.