TITLE: Is there anything like GF(6)?
QUESTION [11 upvotes]: Are there any galois fields which consist of product of two primes, as in $\mathrm{GF}(2\cdot 3) = \mathrm{GF}(6)$?

REPLY [6 votes]: No, because for any reasonable interpretation of the example given, 2 and 3 are zero-divisors (as is 4) and have no multiplicative inverse, so the structure is not a field.
However there are Galois extensions of fields which have degree 6 (the general cubic over the rationals is an example) - and the Galois group then has order 6.