Let be a prime. How do we show that the number of reducible polynomials over of the form is ?
I just know that is a field, and a polynomial of the form is of degree 2, so there is a theorem which says a polynomial of degree 2 or 3 is reducible over a field if and only if it has a root in that field.
So how can I show the total number of polynomials which have a zero in is ? Any help would be appreciated.
The question further asks: "Determine the number of reducible quadratic polynomials over ".
I think again we need to consider the number of distinct expressions of the form .
I know that quadratic polynomials have the form . So this time we need to use the quadratic formula for the roots:
but how could I find the number of reducible polynomials from this formula?
So for the kind there are p different polynomials. Is this correct? or do I need to add another p to this?
and for the kind there are polynomials. I added the p because "a" can be a root.
So in total polynomials?
P.S. Isn't it also possible for the "a" in the first type to be different from the "a" in the second type?