Ring of Gaussian integers

From Commalg
Revision as of 21:07, 29 January 2014 by Vipul (talk | contribs) (Created page with "{{particular curing}} ==Definition== The '''ring of Gaussian integers''' <math>\mathbb{Z}[i]</math> is defined in the following ways: # It is the subring generated by the [...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
This article defines a particular commutative unital ring.
See all particular commutative unital rings

Definition

The ring of Gaussian integers \mathbb{Z}[i] is defined in the following ways:

  1. It is the subring generated by the ring of rational integers and the element i (a square root of -1) in the field of complex numbers.
  2. It is the integral extension \mathbb{Z}[t]/(t^2 + 1) of the ring of rational integers \mathbb{Z}, with the image of the indeterminate t denoted as i.
  3. It is the ring of integers in the number field \mathbb{Q}(i), a quadratic extension of the rationals given as \mathbb{Q}[t]/(t^2 + 1) (with the image of t denoted i).

Ring properties

Property Meaning Satisfied? Explanation
integral domain product of nonzero elements is nonzero Yes Follows from being a subring of \mathbb{C}
Euclidean domain has a Euclidean norm Yes In fact, the standard algebraic norm (which in this case is the same as the square of the complex modulus) is a Euclidean norm. The key geometric fact used is that the distance of any point in \mathbb{C} from the closest point in \mathbb{Z}[i] is less than 1.
principal ideal domain integral domain and every ideal in it is a principal ideal Yes See Euclidean implies PID
unique factorization domain every element has a unique factorization into irreducibles up to units Yes See PID implies UFD
Noetherian domain integral domain Yes Follows from being a PID
Bezout domain Yes Follows from being a PID
Dedekind domain Yes Follows from being a PID