Does a Finite Field of 36 elements exist?
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.