Word Count : 418

---

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 must satisfy (N(a + b\sqrt{-7}) = a^2 + 7b^2 = 1). Now solve (a^2 + 7b^2 = 1) in integers: * ________ ________ _________ ____ ________ _______.
________ ________ ______ ________ __________ ______ ________.
___ __________ ______ _____ ___ _______ ________ _____ _______ _______.
___ _______ _____ ______ ____ _________ _________ ___.
__________ ________ ______ __________ ______ ___ ________.
_________ _________ __________ ______ _________ _______.
______ __________ _____ ______ ______ ____ _____.
_______ _________ _________ ________ _______ _______ _______ ______ ______ _________ __________ ________.
__________ _______ _____ _________ __________ _________ ___ ______.
___ _________ ___ ________ _______ ___.
________ ______ _____ _____ _________ _____ ______ ____ _________ _____.
________ ___ __________ _______ _______ ______ _________ _________ _______ __________.
_________ ___ _______ ________ ___ _________ ___ _______ ________ ______.
_________ ___ _______ ______ _______.
_______ __________ ________ _____ ____ ________ _________.
_______ __________ ______ ____ ________ ________ ___.
_____ _________ __________ __________ ______.
___ ___ _______ __________ ______.
________ _________ __________ _________ _________.
___ _______ _________ _________ _____ ________.
_____ ____ ___ _______ _________ ________.
__________ _____ __________ _________ ________ ____ _________ _____ ___.
________ ________ ________ _________ ____ ___ _____ _____.
_________ ______ __________ _____ _______ ____ ______ ______ ________ _______.
________ _________ _______ ___ _____ _______ _________ ___ __________ _________ ______ _________.
___ ______ _______ ____ ______ _______ _________ _________ _________ _____.
_____ ______ ___ ______ ________ __________ _______ ___ ____ ____ _________.
__________ ________ ______ _____ __________ ____ ___ ____ __________ __________.
_________ _____ _______ ____ _________ _____ ______ ____ __________ ____ ___ _____.
________ _____ _______ ___ _______ ______ _______ ________ _________ __________ _________ ______.
___ ______ __________ ________ ___ ___ __________ _____ __________ _________ ________ _____.
_______ ___ _____ _________ ______ ______.
_______ ______ _______ _______ ______ _______ _________ _____ _______ ____.
____ _______ _________ _______ _____ ___ ___.
_______ _______ ____ ___ __________ _________ ________ _____.
_________ ______ ____ _________ __________ ____ __________ _________.
________ _____ __________ _____ _________ __________ ____ _____ ___.
_________ _________ __________ ___ ___ __________ _______.
Get Full Answer on WhatsApp