Does a Finite Field of 36 elements exist? I'm having a little trouble understanding the finite fields theory, so I'm sorry if my question would seem a little stupid.

I wanted to know if a finite field of 36 elements could exist.
Basically, I thought of the field $\mathbb{F}_{37}$, which has 37 elements. But does for example the invertibles of $\mathbb{F}_{37}$ form a field of 36 elements or not? Since all elements are invertible and we have 36 invertibles in $\mathbb{F}_{37}$.
Finite fields only exist for an order $q$ if $q$ is prime or a power of a prime (http://mathworld.wolfram.com/FiniteField.html). Since 36 is neither a prime itself, nor it is a power of a prime ($36 = 2^3 * 3^2$), there is no finite field with 36 elements.