TITLE: Does a Finite Field with Four Elements Actually Exist?
QUESTION [7 upvotes]: 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.

REPLY [2 votes]: Simply observe that addition and multiplication as defined in the table is equivalent to addition and multiplication of integer polynomials in $x$ modulo $2$ and modulo $x^2+x+1$, and then observe that they commute in a certain sense.
$
\def\zz{\mathbb{Z}}
$
Namely, for any $f,g \in \zz[x]$ define:
  $f \oplus g = (f+g) \bmod 2 \bmod (x^2+x+1)$.
  $f \odot g = (f·g) \bmod 2 \bmod (x^2+x+1)$.
Then prove that for any $f,g,m \in \zz[x]$ we have:
  $f \bmod m \bmod m = f \bmod m$.
  $(f+g) \bmod m = ( f \bmod m + g \bmod m ) \bmod m$.
  $(f·g) \bmod m = ( ( f \bmod m ) · ( g \bmod m ) ) \bmod m$.
Together these can be easily used to show (for you to do!) that associativity and commutativity and distributivity of $\oplus,\odot$ on the set of polynomials $F = \{0,1,x,x+1\}$ is equivalent to associativity and commutativity and distributivity of $+,·$ on integer polynomials, which is very easy to understand intuitively and prove.