dear StackExchange community, so once again I'm confused with this problem concerning finite fields. The given field has four elements with the addition table as
$$\begin{array}{c|c|c|} + & 0 & 1 & x & x + 1 \\ \hline 0 & 0 & 1 & x & x + 1 \\ \hline 1 & 1 & 0 & x + 1 & x \\ \hline x & x & x + 1 & 0 & 1 \\ \hline x + 1 & x + 1 & x & 1 & 0 \\ \hline \end{array}$$
and multiplication table as
$$\begin{array}{c|c|c|} \times & 0 & 1 & x & x + 1 \\ \hline 0 & 0 & 0 & 0 & 0 \\ \hline 1 & 0 & 1 & x & x + 1 \\ \hline x & 0 & x & x + 1 & 1 \\ \hline x + 1 & 0 & x + 1 & 1 & x \\ \hline \end{array}$$
Now the question I'm supposed to answer is "Does a finite field with four elements actually exist?". I know the answer to this question must be yes (via looking at multiple threads here) and I could argue the existence by proving every field axiom, but I don't think that is the wanted solution (proving distribution law would also take forever).
I'm just very confused about the question itself. My course is still very elementary and I don't understand the meaning of a field to exist. Maybe the question is poorly worded? I just don't know. Any help would be nice and thank you in advance.