Which of the following statesment are true?
- there exists a finite field in which additive group is not cyclic
- $F$ is a finite field then there exist a polynomial $p$ over $F$ such that $p(x) \ne0$ for all $x\in F$, where $0$ denotes zeros of $F$
- Every finite field is isomorphic to a subfield of the field of complex numbers
1. is true: Is there a finite field in which the additive group is not cyclic? Actually every finite field other then $\mathbb{Z}_p$ has non-cyclic additive group.
2. is true: the obvious choice is a constant polynomial $p(x)=1$. But there are also such polynomials of positive degree. If $F=\{a_1,\ldots, a_n\}$ then define $$p(x)=(x-a_1)\cdots(x-a_n)+1$$
3. Is false: $\mathbb{C}$ is a field of characteristic $0$. Every finite field has a positive characteristic. Which means that no finite field embeds as a subfield into $\mathbb{C}$.