Characteristic of a field extension of this form

Question

How to prove this is a field?

Following this post, how would one calculate the characteristic of F?

Let $F=(\Bbb Z/5 \Bbb Z)[x]/(x^2+2x+3)$.

Answer 1

Characteristic of a field determines the prime subfield (and vice versa), which is either $\Bbb Q$ or $\Bbb Z/p\Bbb Z$. So which of these is contained in your $F$?

Answer 2

HINT:

Take the identity element in $F$, which is obviously $1$. Keep adding it to itself for a little bit. Eventually it will become $0$ and the number of times you've added $1$ in order to get $0$ is the charactersitic of the field.

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Actually you can see easily see that the characteristic is $p$. Assume that the characteristic is finite and it's $n$. Then as $[n] \in \mathbb{Z}_p$ we have that $[n][1] = [0]$, but as we're working in a field we must have $[n] = [0]$ or $[1] = [0]$. The latter is impossible, so we have that $[n] = [0]$. Now the charactersitic is the smallest positive integer of $[0]$, which is $p$.