I'm learning cryptography and today I learned Galoi's Field, but I cannot understand why $\Bbb Z_p[x]$ is finite field.
I know $\Bbb Z_p$ is finite filed because $p$ is prime number and by EEA, but $Z_p[x]$ has too many polynomials which degree is $x^0$ ~ $x^{\infty}$.
To understand Galoi's field, I should understand why that is finite field. How can I prove it?
$\Bbb Z_p[x]$ is not a field because $x$ is not invertible.
$\Bbb Z_p[x]$ is not finite because $x^n$ for $n\in\Bbb N$ are infinitely many different elements.