Word Count : 418

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The ring (\mathbb{Z}[\sqrt{-7}]) consists of elements of the form (a + b\sqrt{-7}), where (a, b \in \mathbb{Z}). An element (u = a + b\sqrt{-7}) is a unit if it has a multiplicative inverse in the ring. This is equivalent to saying there exists (v = c + d\sqrt{-7} \in \mathbb{Z}[\sqrt{-7}]) such that (uv = 1). Compute the norm (N(a + b\sqrt{-7}) = a^2 + 7b^2). The key property: if (uv = 1), then (N(u)N(v) = N(uv) = N(1) = 1). So the norm _______ _________ ______ ____ _______ _______ ____ _____ __________ _____ ____.
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