What does the homomorphism $\rho: \mathbb Z[X] \longrightarrow \mathbb R$, $X\longmapsto 1+\sqrt2$ describe?

Question

Is it the set of all polynomials with integer coefficients, where one then "plugs in" $1+\sqrt2$?

For example: $$a_nX^n+a_{n-1}X^{n-1}+\cdots+a_1X+a_0$$ becomes $$a_n(1+\sqrt2)^n+a_{n-1}(1+\sqrt2)^{n-1}+\cdots+a_1(1+\sqrt2)+a_0$$

or something different?

Answer 1

Yes, what you've described is the correct interpretation.