Word Count : 431

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Let's carefully analyze this problem step by step. We are asked: > Let > > $$ > D = \{ f(x,y) + g(x,y)i \mid f,g \in \mathbb{Z}[x,y] \} \subseteq \mathbb{C}[x,y]. > $$ > > Check whether $D$ is a UFD (Unique Factorization Domain) or not. --- ### Step 1: Understanding the structure of $D$ 1. Elements of $D$ are of the form $f(x,y) + i g(x,y)$, where $f, g$ are polynomials with integer coefficients in variables $x$ and $y$. 2. So, $D = \mathbb{Z}[x,y] + i __________ ______ _______ __________ _________.
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