<s>[INST] Question about constructing fields 
Find all the monic irreducible polynomials in $F_5[x]$ of degree two (aside
  from $x^2-2$ and $x^2-3$, there are eight of them) Adjoining a root u of
  these polynomials to $F_5$, construct eight fields $F_5(u)$ of $25$ elements.
  Prove that each of these fields is isomorphic to $F_5(\sqrt 2)^\times$

Apart from $x^2-2$ and $x^2-3$, I've found that there are $8$ monic irreducible polynomials in $F_5[x]$, which are:
$x^2+x+1,\,\, x^2+x+2,\,\, x^2+2x+3,\,\, x^2+2x+4,\,\, x^2+3x+3,\,\, x^2+3x+4,\,\, x^2+4x+1,\,\, x^2+4x+2$
For example for the polynomial $x^2+x+1$ we have
$F_5[u]=F_5[x]/x^2+x+1$  where we identify $u$ with the image of $x $ in $F_5[u]$ 
Now, how to construct such a field of $25$ elements and show that this field is isomorphic to $F_5(\sqrt 2)^\times$?
 [/INST] For example, let's see that there is a root of $x^2 + x + 1$ in $F_5[\sqrt{2}]$.
If $\alpha^2 = 2$ and $\beta = x \alpha + y$ we have
$$\beta^2 + \beta + 1 = (2 x^2 + y^2 + y + 1) + (2 x y + x)\alpha = 0$$ if $x=y=2$.  So we get an isomorphism from $F_5[x]/(x^2+x+1)$ into $F_5[\sqrt{2}]$.  Since the cardinalities are equal it's also onto.
</s>