How to show if it is irreducible

Question

Let R be the subring (with identity) of Q[x, y] generated by x^2 , y^2 , and xy. Each of these elements is irreducible in R

How do we know if they are irreducible

Answer 1

A non-unit element is irreducible if it cannot be written as the product of two non-unit elements. Up to association, $x^2=x\cdot x$ is the only factorization of $x^2$, but $x\notin R$, so $x^2$ is irreducible. Similarly, $y^2$ uniquely factorizes as $y\cdot y$, but $y\notin R$, and $xy=x\cdot y$ and $x,y\notin R$.

Answer 2

Under $\mathbb Q[x,y]\to\mathbb Q[t]$, $x\mapsto t, y\mapsto t$, all your candidates map to $t^2$, and the subring of $\mathbb Q[t]$ generated by $t^2$ is isomorphic to $\mathbb Q[u]$ via $u\mapsto t^2$. Finally $u\in\mathbb Q[u]$ is clearly irreducible.