Question

Prove that x 2 + y2 - 1 is irreducible in q[ x, y].

Answer 1

Suppose that the polynomial x^2+y^2-1 is not irreducible. 
It means that there exists two fractions such that :
$(\frac{p}{q})^2+( \frac{s}{t} )^2 -1=0\\\text{rearranging the above formula we get:}\\p^2t^2=q^2(t^2-s^2)\\\text{since q and p are primes each other, we deduce that }\\\text{q divide t meaning that we can write }t=a\times t\text{ for an integer a}\\\text{rearranging again the formula and using our finding we get:}\\q^2s^2=a^2q^2(q^2-p^2)\\\text{Same argument as above, we deduce that q divide s.}$
We found that q divide t and q divide s, but s and t are supposed to be prime each other. It is a contradiction. 
The polynomial x^2+y^2-1 is then irreducible in q[x,y]