Advances in Commutative Algebra pp 171-195 | Cite as

# How Do Elements Really Factor in \(\mathbb {Z}[\sqrt{-5}]\)?

## Abstract

Most undergraduate level abstract algebra texts use \(\mathbb {Z}[\sqrt{-5}]\) as an example of an integral domain which is not a unique factorization domain (or UFD) by exhibiting two distinct irreducible factorizations of a nonzero element. But such a brief example, which requires merely an understanding of basic norms, only scratches the surface of how elements actually factor in this ring of algebraic integers. We offer here an interactive framework which shows that while \(\mathbb {Z}[\sqrt{-5}]\) is not a UFD, it does satisfy a slightly weaker factorization condition, known as half-factoriality. The arguments involved revolve around the Fundamental Theorem of Ideal Theory in algebraic number fields.

## 2010 AMS Mathematics Subject Classification

Primary 11R11 13F15 20M13## Notes

### Acknowledgements

It is a pleasure for the authors to thank the referee, whose helpful suggestions vastly improved the final version of this paper.

## References

- 1.L. Carlitz, A characterization of algebraic number fields with class number two. Proc. Am. Math. Soc.
**11**, 391–392 (1960)MathSciNetzbMATHGoogle Scholar - 2.S.T. Chapman, A tale of two monoids: A friendly introduction to nonunique factorizations. Math. Mag.
**87**, 163–173 (2014)MathSciNetCrossRefGoogle Scholar - 3.S.T. Chapman, So what is class number 2? Am. Math. Mon. to appearGoogle Scholar
- 4.S.T. Chapman, J. Herr, N. Rooney, A factorization formula for class number two. J. Number Theory
**79**, 58–66 (1999)MathSciNetCrossRefGoogle Scholar - 5.S.T. Chapman, U. Krause, E. Oeljeklaus, Monoids determined by a homogeneous linear diophantine equation and the half-factorial property. J. Pure Appl. Algebra
**151**, 107–133 (2000)MathSciNetCrossRefGoogle Scholar - 6.F. Jarvis,
*Algebraic Number Theory*, Springer Undergraduate Mathematics Series (Springer, New York, 2014)zbMATHGoogle Scholar - 7.H. Kim, Examples of half-factorial domains. Canad. Math. Bull.
**43**, 362–367 (2000)MathSciNetCrossRefGoogle Scholar - 8.D. Marcus,
*Number Fields*(Springer, New York, 1977)CrossRefGoogle Scholar - 9.H. Pollard, H.G. Diamond,
*The Theory of Algebraic Numbers*(Courier Corporation, New York, 1998)Google Scholar - 10.M. Ram Murty, J. Esmonde,
*Problems in Algebraic Number Theory*, vol. 190, Graduate Text in Mathematics (Springer, New York, 2005)zbMATHGoogle Scholar