# Is there anything like GF(6)?

Are there any galois fields which consist of product of two primes, as in $$\mathrm{GF}(2\cdot 3) = \mathrm{GF}(6)$$?

No; finite fields must have order a power of prime, and for every prime $p$ and every $n\gt 0$, there is one and only one (up to isomorphism) field of order $p^n$.
To see why the order must be the power of a prime: note that the characteristic of an integral domain must be a prime; since a field is an integral domain, it must be of characteristic $p$ for some prime $p$. It is now easy to see that the field is in fact a vector space over the field of order $p$; since it is finite, it must have a basis with $n$ elements; but a vector space of dimension $n$ over a field with $p$ elements must have $p^n$ vectors.
To see why there is one and only one up to isomorphism, consider the splitting field of $x^{p^n}-x$ over $GF(p)$.