Number of irreducible quadratic polynomials over a finite field To  find  the  number  of  irreducible  polynomials of  the  form $x^{2} + ax+b$  over  the  field  $\Bbb{F}_{7}$  I  manually  checked  all  the  possibilities  and  thus  found  the  answer  to  be  $21.$ Like $b=0$  is  not  possible  or  taking  $b=1$  and  checking  what  values  of $a$  works  here  etc. Surely it  was   tedious  and  time-consuming  process  and  I  don't  think  it  was  appropriate  either. For  what  should  be  done for  a much  larger  field$?$  Are  there  some  basic  theories  that  I  should  use. Hints  please.
 A: The number of monic quadratic polynomials over $\Bbb{F}_q$ is $q^2$; they are the polynomials
$$X^2+aX+b,$$
with $a,b\in\Bbb{F}_q$. If a monic quadratic polynomial $P\in\Bbb{F}_q[X]$ is reducible then
$$P(X)=(X-a)(X-b),$$
for some $a,b\in\Bbb{F}_q$. So how many reducible monic quadratic polynomials are there?

 There are $\tfrac{1}{2}q(q+1)$ reducible monic quadratic polynomials in $\Bbb{F}_q$.
 This yields a total of $\tfrac{1}{2}q(q-1)$ irreducible monic quadratic polynomials in $\Bbb{F}_q[X]$. For $q=7$ we indeed get $21$ such polynomials.

A: Regardless of the  field, if  a monic polynomial $$x^2+ax+b$$ is given then it can be reducible in two different ways:
$$(x-p)(x-q);p\neq q\\\text{or}\\(x-p)(x-q)={(x-p)}^2;p=q$$
If the  field   has  cardinality  $N$, then the total  number  of  quadratic  monic  polynomials  is  $N^2$.  There  are $\binom{N}{2}$ (Combination without repetition)  ways  of  choosing $\{p,q\}$ in the first case and $N$  ways in the second. We can conclude that the number  of reducible  polynomials  is  $${\binom{N}{2}}+N\\={{{N(N+1)}}\over 2}$$
So  the  number  of  irreducible polynomials  is  $$N^2-{{{N(N+1)}}\over 2}\\=N\cdot{{N-1}\over 2}$$
Since $N$ was arbitrary the claim follows.
A: In any field (characteristic $\neq 2$) the quadratic formula still holds.  The equation: $x^2 + ax + b = 0$ has two solutions:
$$ x = \frac{-a \pm \sqrt{a^2 - 4b}}{2}$$
Then as $(a,b) \in \mathbb{F}_7^2$ how many of these have $a^2 - 4b$ a quadratic residue?  Since $4$ is invertible mod $7$
$$ a^2 - 4 \,\mathbb{F}_7 = \mathbb{F}_7$$
If we replace $7$ with a large prime $p \gg 1$ the odds of being a quadratic residue approach $\frac{1}{2}$.  

What does the quadratic formula look like in characteristic 2?  In $\mathbb{F}_2$ it is easy enough to check by hand.  There are also finite fields with $2^n$ elements such as $\mathbb{F}_8$ which are also characteristic 2.
See also: Number of monic irreducible polynomials of prime degree $p$ over finite fields
