I am trying to answer the question already asked here.
My question is two parts:
1) I think I have found a proof on my own, could someone check it is valid?
Modulo that ideal, $x_i\equiv a_i$ so any polynomial in $k[x_1,\cdots, x_n]$ is congruent to the polynomial evaluated at $(a_1,\cdots,a_n)$ which is just an element of $k.$ Hence non-zero elements are invertible, so the quotient is a field and the ideal is maximal.
2) Trying to following DonAntonio's answer, I can't actually compute the kernel of that homomorphism. I can show that ideal is in the kernel, but not the reverse inclusion. Any hints?