Cardinality of a field between interval

Question

I was studying for an exam when I came across the following question:

Let $F$ a field such that $4<|F|<15$. The number of elements in $F$ is:
A. 6 or 7
B. 11 or 13
C. 8 or 9
D. 7 or 14

For better understanding of the problem, I defined $|x|=|F|, x \in \mathbb{R}$

Then, after some calculations, I found that $|x|=11$, but I can't understand why there is 2 possible sizes for this field.

Why would the cardinality of $F$ be two different numbers? Is it possible?

Answer 1

The question is asking what cardinalities are possible for $F$ to have without any further information being given. The reason for this is that a finite field has as its order the power of some prime number, i.e., $p^n$ for some prime $p$ and $n\geq 1$. The only such numbers between $4$ and $15$ are those in options B and C.

However, all options listed in B and C are possible: there exists a finite field of size $8$, but also one of $9$, or one of $11$, or one of size $13$ (actually, only one for each size, up to isomorphism).

So I think this is a bad question, unless multiple answers were allowed...

Answer 2

With the limited information given, we cannot conclude the precise order of $F $. A useful hint is that the order of a finite field $F$ is $p^n $ ,where $p$ is the characteristic of $F$(and it's a prime).