(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Kernel is a principle ideal?

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