Example 1. Determine whether the polynomial $\mathrm{x}^{2}+(\sqrt{2}-1) \mathrm{xy}$
$$
-\sqrt{2} y^{2}+\frac{\sqrt{2}}{2} \mathrm{x}-(2 \sqrt{2}+1) y-1 \text { can be factored over }
$$

the field of characteristic number.
$$
\begin{array}{l}
\text { Solution: Since } \Delta^{\prime}=\left[(\sqrt{2}-1) \cdot \frac{\sqrt{2}}{2}\right. \\
+2(2 \sqrt{ } 2+1)]^{2}-\left[(\sqrt{2}-1)^{2}+4 \sqrt{2}\right] \\
{\left[\left(\frac{\sqrt{2}}{2}\right)^{2}+4\right]=20+12 \sqrt{2} \neq 0,}
\end{array}
$$

Therefore, the original polynomial cannot be factored over the complex numbers.