What is $\mathbb Z[t]/(t,5)$?

Question

What is $\mathbb Z[t]/(t,5)$ ?

with ad-hoc thinking, I would say that we have $$\mathbb Z[t]/(5,t)\cong \mathbb F_5[t]/(t)\cong \mathbb F_5,$$ but I don't know why. It's a multiple choice question, and since it's not in the proposition It's probably wrong, but the choice are :

1) $\mathbb Z[t]/(t+5)$

2) $\mathbb F_5\times \mathbb Z$

3) $\mathbb F_5[t]$

4) $\mathbb Z[t]$

But no proposition looks good. I can't prove that it's $\mathbb F_5$, but I think it's correct, but I can't prove it. Any help would be welcome.

Answer 1

Using 3rd isomorphism theorem $$\mathbb Z[t]/(5,t)\cong (\mathbb Z[t]/(t))/((5,t)/(t))$$ Now, $$\mathbb Z[t]/(t)\cong \mathbb Z$$ and $$(5,t)/(t)\cong (5).$$ Therefore $$\mathbb Z[t]/(5,t)\cong \mathbb Z/(5)=\mathbb F_5.$$

Answer 2

we have $\mathbb Z[x]/(p,f(x)) \cong \mathbb Z_p[x]/(f(x)) $, in question we have $\mathbb Z[t]/(5,t)) \cong \mathbb Z_5[t]/(t) \cong \mathbb Z_5 $