So I need to show that $\mathbb F_2[x,y]/(x-1,y-1)$ is an integral domain.
If I define a homomorphism $\varphi:\mathbb F_2[x,y]\to\mathbb F_2[x,y]/(x-1,y-1)$ the kernel of the homomorphism is $(x-1,y-1)$. So what's next ? I am clueless.
2026-03-30 15:17:15.1774883835
Prove that $(x-1,y-1)$ is a prime ideal in $\mathbb F_2[x,y]$
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Hint:
Consider the ring homomorphism \begin{align} \mathbf F_2[x,y]&\longrightarrow \mathbf F_2 ,\\ p(x,y)&\longmapsto p(1,1), \end{align} check it is surjective and determine its kernel.