bibkey: chatland1949euclidean authors: H. Chatland year: 1949 title: On the Euclidean algorithm in quadratic number fields doi: 10.1090/S0002-9904-1949-09315-1 claim: The ring of integers of Q(sqrt(5)) admits Euclidean division for the absolute field norm. strata_touched:

• D5/S0/Carrier/Euclidean license: citation-only triage: anchor

On the Euclidean Algorithm in Quadratic Number Fields

H. Chatland defines a quadratic field as Euclidean when, for algebraic integers alpha and nonzero beta, there is an algebraic integer gamma satisfying |N(alpha - beta * gamma)| < |N(beta)|. The paper explicitly lists m = 5 among the positive square-free values for which this algorithm is known to exist. The repository’s GoldenInt is the integral-basis model of this ring of integers, Z[phi].

The paper attests the theorem and its norm inequality. The deterministic nearest-coordinate quotient, the 5/16 estimate, and the exact Lean EuclideanDomain construction are the repository’s formal proof choices.

Search log

• Queried Crossref for The Euclidean algorithm in algebraic number fields. Crossref returned Chatland’s title, author, year, and DOI 10.1090/S0002-9904-1949-09315-1.
• Downloaded the six-page article from the AMS journal archive, rendered its first page, and checked both the page image and extracted text. The introduction gives the strict absolute-norm remainder inequality, and the previous-results section includes 5 in the positive Euclidean list.

Verified locator

• DOI: https://doi.org/10.1090/S0002-9904-1949-09315-1
• AMS PDF: https://www.ams.org/journals/bull/1949-55-10/S0002-9904-1949-09315-1/S0002-9904-1949-09315-1.pdf