Show that the polynomial $x^2+y^2-1$ is irreducible over $\mathbb{Q}[x,y]$ 
Show that the polynomial $x^2+y^2-1$ is irreducible over $\mathbb{Q}[x,y]$. 

Theorem : Let $A$ an integral domain and $I$ a proper ideal of $A$. If $f(x) \not \equiv a(x)b(x) \pmod I$ for any polynomials $a(x)$, $b(x)$ $\in A[x]$ of degree $\in [1, \deg(f))$, then $f(x)$ is irreducible in $A[x]$
I think I have to use this theorem, maybe in using the ideal $(y^2)$
The fact that $\mathbb{Q}[x,y]=\mathbb{Q}[x][y]$ and $\mathbb{\mathbb{Q}[y]}$ is a unique factorization domain could it help me here? Is anyone could help me at this point?

As you observed, $\mathbf{Q}[x,y] = \mathbf{Q}[x][y]$ so viewing $x^2 + y^2 - 1$ as a polynomial with coefficients in $\mathbf{Q}[x]$, we can apply Eisenstein with, for example, $(x+1)$, which is a prime ideal in $\mathbf{Q}[x]$. 
