This paper introduces a totally real quartic number field with a non-principal Euclidean ideal.

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Lenstra's concept of Euclidean ideals generalizes the Euclidean algorithm; a domain with a Euclidean ideal has cyclic class group, while a domain with a Euclidean algorithm has trivial class group. This paper generalizes Harper's variation of Motzkin's lemma to Lenstra's concept of Euclidean ideals and then uses the large sieve to obtain growth results. It concludes that if a certain set of primes is large enough, then the ring of integers of a number field with cyclic class group has a Euclidean ideal.

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In 1949, Motzkin proved that every Euclidean domain R has a minimal Euclidean function, φ R . He showed that when R = Z, the minimal function is φ Z (x) = ⌊log 2 |x|⌋. For over seventy years, φ Z has been the only example of an explictly-computed minimal function in a number field. We give the first explicitly-computed minimal function in a non-trivial number field, φ Z[i] , which computes the length of the shortest possible (1 + i)-ary expansion of any Gaussian integer. We also present an algorithm that uses φ Z[i] to compute minimal (1 + i)-ary expansions of Gaussian integers. We solve these problems using only elementary methods.