A surjective ring homomorphism I am reading P. E. Frenkel, Simple proof of Chebotarev's theorem on roots of unity. The proof of Lemma 1 claims the following.


*

*Function
$$Z[\Omega] \rightarrow Z[\Omega]/(1 − \Omega, p) = F_p,\quad \Omega \rightarrow 1$$ is a surjective ring homomorphism.


*Therefore, the latter homomorphism factors through the former one via a surjective homomorphism $Z[ω]\rightarrow F_p$ whose kernel is
the ideal $$(1 − \Omega, p)/(\Phi_p(\Omega)) = (1−\omega, p) = (1−ω),$$

I do not see these equalities, and the surjectivity is thus not obvious either. I do not know what $(\cdot, \cdot)$ here means. Is $\Omega\rightarrow 1$ supposed to be an evaluation homomorphism?
I am sure it is obvious to someone who is versed in abstract algebra. I would appreciate the elucidation.
 A: $(a,b)$ is the ideal generated by $a$ and $b$.

*

*It is just a composite of two surjective morphisms:


*

*the evaluation ${\bf Z}[\Omega]\to{\bf Z}[\Omega]/(1−\Omega)={\bf Z},\quad \Omega\mapsto1$;

*the "mod p" morphism: ${\bf Z}\to{\bf Z}/(p)={\bf F}_p$.



*


*

*The first equality is a restriction (to ideals) of the equality of rings ${\bf Z}[\Omega]/(1+\Omega+\dots+\Omega^{p-1})={\bf Z}[\omega]$ which constitutes the definition of $\omega$.

*The second equality is because modulo $1-\omega$, we have $\omega^k\equiv1$ ($\forall k$), hence $$p=1+1+\dots+1\equiv1+\omega+\dots+\omega^{p-1}=0,$$so $p\in(1-\omega)$.

*The fact that the morphism $w:{\bf Z}[\Omega]\to{\bf F}_p$ (of 1.) "factors" through the canonical morphism $u:{\bf Z}[\Omega]\to{\bf Z}[\omega]$ via some morphism $v:{\bf Z}[\omega]\to{\bf F}_p$ (i.e. $w=v\circ u$) is due to the inclusion $\ker u\subset\ker w$, i.e. $$(1+\Omega+\dots+\Omega^{p-1})\subset(1−\Omega,p).$$ This is because $$p\equiv1+\Omega+\dots+\Omega^{p-1}\bmod{1-\Omega}$$ (by the same trick as above for $\omega$).

*This new morphism $v$ is surjective because the morphism $v\circ u=w$ is.

*Finally, since $u$ is surjective, $\ker v=u(\ker w)=u(1−\Omega,p)=(1−\Omega,p)/(1+\Omega+\dots+\Omega^{p-1}).$
