Ring of integers
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In mathematics, the ring of integers of an algebraic number field is the ring of all algebraic integers contained in .: .[2] This ring is often denoted by or . Since any integer belongs to and is an integral element of , the ring is always a subring of .
The ring of integers is the simplest possible ring of integers.[a] Namely, where is the field of rational numbers.[3] And indeed, in algebraic number theory the elements of are often called the "rational integers" because of this.
The next simplest example is the ring of Gaussian integers , consisting of
The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain.[4]
Properties
The ring of integers OK is a
with ai ∈ Z. of K over Q.
Examples
Computational tool
A useful tool for computing the integral closure of the ring of integers in an algebraic field K/Q is the discriminant. If K is of degree n over Q, and form a basis of K over Q, set . Then, is a
Cyclotomic extensions
If p is a prime, ζ is a pth root of unity and K = Q(ζ ) is the corresponding cyclotomic field, then an integral basis of OK = Z[ζ] is given by (1, ζ, ζ 2, ..., ζ p−2).[7]
Quadratic extensions
If is a square-free integer and is the corresponding quadratic field, then is a ring of quadratic integers and its integral basis is given by (1, (1 + √d) /2) if d ≡ 1 (mod 4) and by (1, √d) if d ≡ 2, 3 (mod 4).[8] This can be found by computing the minimal polynomial of an arbitrary element where .
Multiplicative structure
In a ring of integers, every element has a factorization into
A ring of integers is always a Dedekind domain, and so has unique factorization of ideals into prime ideals.[10]
The
Generalization
One defines the ring of integers of a
For example, the
Qp .See also
- Minimal polynomial (field theory)
- Integral closure– gives a technique for computing integral closures
Notes
Citations
- ^ Alaca & Williams 2003, p. 110, Defs. 6.1.2-3.
- ^ Alaca & Williams 2003, p. 74, Defs. 4.1.1-2.
- ^ a b Cassels 1986, p. 192.
- ^ a b Samuel 1972, p. 49.
- ^ Cassels (1986) p. 193
- ^ a b Baker. "Algebraic Number Theory" (PDF). pp. 33–35.
- ^ Samuel 1972, p. 43.
- ^ Samuel 1972, p. 35.
- ISBN 978-0-13-241377-0.
- ^ Samuel 1972, p. 50.
- ^ Samuel 1972, pp. 59–62.
- ^ Cassels 1986, p. 41.
References
- Alaca, Saban; Williams, Kenneth S. (2003). Introductory Algebraic Number Theory. ISBN 9780511791260.
- Zbl 0595.12006.
- Zbl 0956.11021.
- Samuel, Pierre (1972). Algebraic number theory. Hermann/Kershaw.