fbpx
Wikipedia

Ring of integers

In mathematics, the ring of integers of an algebraic number field is the ring of all algebraic integers contained in .[1] An algebraic integer is a root of a monic polynomial with integer coefficients: .[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 complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, is a Euclidean domain.

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 edit

The ring of integers OK is a finitely-generated Z-module. Indeed, it is a free Z-module, and thus has an integral basis, that is a basis b1, ..., bn ∈ OK of the Q-vector space K such that each element x in OK can be uniquely represented as

 

with aiZ.[5] The rank n of OK as a free Z-module is equal to the degree of K over Q.

Examples edit

Computational tool edit

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 submodule of the Z-module spanned by  .[6] pg. 33 In fact, if d is square-free, then   forms an integral basis for  .[6] pg. 35

Cyclotomic extensions edit

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, ..., ζp−2).[7]

Quadratic extensions edit

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 edit

In a ring of integers, every element has a factorization into irreducible elements, but the ring need not have the property of unique factorization: for example, in the ring of integers Z[−5], the element 6 has two essentially different factorizations into irreducibles:[4][9]

 

A ring of integers is always a Dedekind domain, and so has unique factorization of ideals into prime ideals.[10]

The units of a ring of integers OK is a finitely generated abelian group by Dirichlet's unit theorem. The torsion subgroup consists of the roots of unity of K. A set of torsion-free generators is called a set of fundamental units.[11]

Generalization edit

One defines the ring of integers of a non-archimedean local field F as the set of all elements of F with absolute value ≤ 1; this is a ring because of the strong triangle inequality.[12] If F is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers. The ring of integers of an algebraic number field may be characterised as the elements which are integers in every non-archimedean completion.[3]

For example, the p-adic integers Zp are the ring of integers of the p-adic numbers Qp.

See also edit

Notes edit

  1. ^ The ring of integers, without specifying the field, refers to the ring   of "ordinary" integers, the prototypical object for all those rings. It is a consequence of the ambiguity of the word "integer" in abstract algebra.

Citations edit

  1. ^ Alaca & Williams 2003, p. 110, Defs. 6.1.2-3.
  2. ^ Alaca & Williams 2003, p. 74, Defs. 4.1.1-2.
  3. ^ a b Cassels 1986, p. 192.
  4. ^ a b Samuel 1972, p. 49.
  5. ^ Cassels (1986) p. 193
  6. ^ a b Baker. "Algebraic Number Theory" (PDF). pp. 33–35.
  7. ^ Samuel 1972, p. 43.
  8. ^ Samuel 1972, p. 35.
  9. ^ Artin, Michael (2011). Algebra. Prentice Hall. p. 360. ISBN 978-0-13-241377-0.
  10. ^ Samuel 1972, p. 50.
  11. ^ Samuel 1972, pp. 59–62.
  12. ^ Cassels 1986, p. 41.


References edit

ring, integers, mathematics, ring, integers, algebraic, number, field, displaystyle, ring, algebraic, integers, contained, displaystyle, algebraic, integer, root, monic, polynomial, with, integer, coefficients, displaystyle, cdots, this, ring, often, denoted, . In mathematics the ring of integers of an algebraic number field K displaystyle K is the ring of all algebraic integers contained in K displaystyle K 1 An algebraic integer is a root of a monic polynomial with integer coefficients xn cn 1xn 1 c0 displaystyle x n c n 1 x n 1 cdots c 0 2 This ring is often denoted by OK displaystyle O K or OK displaystyle mathcal O K Since any integer belongs to K displaystyle K and is an integral element of K displaystyle K the ring Z displaystyle mathbb Z is always a subring of OK displaystyle O K The ring of integers Z displaystyle mathbb Z is the simplest possible ring of integers a Namely Z OQ displaystyle mathbb Z O mathbb Q where Q displaystyle mathbb Q is the field of rational numbers 3 And indeed in algebraic number theory the elements of Z displaystyle mathbb Z are often called the rational integers because of this The next simplest example is the ring of Gaussian integers Z i displaystyle mathbb Z i consisting of complex numbers whose real and imaginary parts are integers It is the ring of integers in the number field Q i displaystyle mathbb Q i of Gaussian rationals consisting of complex numbers whose real and imaginary parts are rational numbers Like the rational integers Z i displaystyle mathbb Z i is a Euclidean domain The ring of integers of an algebraic number field is the unique maximal order in the field It is always a Dedekind domain 4 Contents 1 Properties 2 Examples 2 1 Computational tool 2 2 Cyclotomic extensions 2 3 Quadratic extensions 3 Multiplicative structure 4 Generalization 5 See also 6 Notes 7 Citations 8 ReferencesProperties editThe ring of integers OK is a finitely generated Z module Indeed it is a free Z module and thus has an integral basis that is a basis b1 bn OK of the Q vector space K such that each element x in OK can be uniquely represented as x i 1naibi displaystyle x sum i 1 n a i b i nbsp with ai Z 5 The rank n of OK as a free Z module is equal to the degree of K over Q Examples editComputational tool edit 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 a1 an OK displaystyle alpha 1 ldots alpha n in mathcal O K nbsp form a basis of K over Q set d DK Q a1 an displaystyle d Delta K mathbb Q alpha 1 ldots alpha n nbsp Then OK displaystyle mathcal O K nbsp is a submodule of the Z module spanned by a1 d an d displaystyle alpha 1 d ldots alpha n d nbsp 6 pg 33 In fact if d is square free then a1 an displaystyle alpha 1 ldots alpha n nbsp forms an integral basis for OK displaystyle mathcal O K nbsp 6 pg 35 Cyclotomic extensions edit If p is a prime z is a p th root of unity and K Q z is the corresponding cyclotomic field then an integral basis of OK Z z is given by 1 z z 2 z p 2 7 Quadratic extensions edit If d displaystyle d nbsp is a square free integer and K Q d displaystyle K mathbb Q sqrt d nbsp is the corresponding quadratic field then OK displaystyle mathcal O K nbsp 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 a bd Q d displaystyle a b sqrt d in mathbf Q sqrt d nbsp where a b Q displaystyle a b in mathbf Q nbsp Multiplicative structure editIn a ring of integers every element has a factorization into irreducible elements but the ring need not have the property of unique factorization for example in the ring of integers Z 5 the element 6 has two essentially different factorizations into irreducibles 4 9 6 2 3 1 5 1 5 displaystyle 6 2 cdot 3 1 sqrt 5 1 sqrt 5 nbsp A ring of integers is always a Dedekind domain and so has unique factorization of ideals into prime ideals 10 The units of a ring of integers OK is a finitely generated abelian group by Dirichlet s unit theorem The torsion subgroup consists of the roots of unity of K A set of torsion free generators is called a set of fundamental units 11 Generalization editOne defines the ring of integers of a non archimedean local field F as the set of all elements of F with absolute value 1 this is a ring because of the strong triangle inequality 12 If F is the completion of an algebraic number field its ring of integers is the completion of the latter s ring of integers The ring of integers of an algebraic number field may be characterised as the elements which are integers in every non archimedean completion 3 For example the p adic integers Zp are the ring of integers of the p adic numbers Qp See also editMinimal polynomial field theory Integral closure gives a technique for computing integral closuresNotes edit The ring of integers without specifying the field refers to the ring Z displaystyle mathbb Z nbsp of ordinary integers the prototypical object for all those rings It is a consequence of the ambiguity of the word integer in abstract algebra Citations edit Alaca amp Williams 2003 p 110 Defs 6 1 2 3 Alaca amp 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 Artin Michael 2011 Algebra Prentice Hall p 360 ISBN 978 0 13 241377 0 Samuel 1972 p 50 Samuel 1972 pp 59 62 Cassels 1986 p 41 References editAlaca Saban Williams Kenneth S 2003 Introductory Algebraic Number Theory Cambridge University Press ISBN 9780511791260 Cassels J W S 1986 Local fields London Mathematical Society Student Texts Vol 3 Cambridge Cambridge University Press ISBN 0 521 31525 5 Zbl 0595 12006 Neukirch Jurgen 1999 Algebraische Zahlentheorie Grundlehren der mathematischen Wissenschaften Vol 322 Berlin Springer Verlag ISBN 978 3 540 65399 8 MR 1697859 Zbl 0956 11021 Samuel Pierre 1972 Algebraic number theory Hermann Kershaw Retrieved from https en wikipedia org w index php title Ring of integers amp oldid 1197291870, wikipedia, wiki, book, books, library,

article

, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.