Euclidean domain - Citizendia

In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm applies. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Number theory, the Euclidean algorithm (also called Euclid's algorithm) is an Algorithm to determine the Greatest common divisor (GCD

A Euclidean domain is a specific type of integral domain, and can be characterized by the following (not necessarily exhaustive) chain of class inclusions:

integral domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields

1 Definition
2 Examples
3 Properties
4 See also
5 References

Definition

Formally, a Euclidean domain is an integral domain D on which one can define a function v mapping nonzero elements of D to non-negative integers that satisfies the following division-with-remainder property:

If a and b are in D and b is nonzero, then there are q and r in D such that a = bq + r and either r = 0 or v(r) < v(b). In Set theory and its applications throughout Mathematics, a subclass is a class contained in some other class in the same way that a Subset In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Mathematics, a unique factorization domain (UFD is roughly speaking a Commutative ring in which every element with special exceptions can be uniquely written In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

The function v is called a valuation or norm or gauge and the key point here is that the remainder r has v-size smaller than the v-size of the divisor b. The operation mapping (a, b) to (q, r) is called the Euclidean division, whereas q is called the Euclidean quotient. In its simplest meaning in Mathematics and Logic, an operation is an action or procedure which produces a new value from one or more input values

Nearly all algebra textbooks which discuss Euclidean domains include the following extra property in the definition:

for all nonzero a and b in D, v(a) ≤ v(ab).

This property does not have to be assumed since it is not needed to prove the most basic facts about Euclidean domains (see below). However, this inequality can always be arranged to occur by changing the choice of v, as follows: if (D,v) is a Euclidean domain as given above then the function w defined on nonzero elements of D by w(a) = least value of v(ax) as x runs over nonzero elements of D also makes D a Euclidean domain according to the above definition and it satisfies w(a) ≤ w(ab) for all nonzero a and b in D.

To check that w is a norm, suppose that b does not divide a and, amongst all expressions of the form a = bq + r, choose one for which v(r) is minimal. If w(r) ≥ w(b), then v(r) ≥v(bc) for some c. We can write a = bcQ + R with v(R) < v(bc) ≤ v(r), which contradicts the minimality of v(r).

Examples

Examples of Euclidean domains include:

Z, the ring of integers. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Define v(n) = |n|, the absolute value of n. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.
Z[i], the ring of Gaussian integers. A Gaussian integer is a Complex number whose real and imaginary part are both Integers The Gaussian integers with ordinary addition and multiplication of complex Define v(a+bi) = a²+b², the norm of the Gaussian integer a+bi.
Z[ω] (where ω is a cube root of 1), the ring of Eisenstein integers. In Mathematics, Eisenstein integers, named after Ferdinand Eisenstein, are Complex numbers of the form z = a + b\omega \\! Define v(a+bω) = a²-ab+b², the norm of the Eisenstein integer a+bω.
K[X], the ring of polynomials over a field K. In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division For each nonzero polynomial f, define v(f) to be the degree of f.
K[[X]], the ring of formal power series over the field K. In Mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of Power series in settings that do not For each nonzero power series f, define v(f) as the degree of the smallest power of X occurring in f.
Any discrete valuation ring. In Abstract algebra, a discrete valuation ring ( DVR) is a Principal ideal domain (PID with exactly one non-zero Maximal ideal. Define v(x) to be the highest power of the maximal ideal M containing x (equivalently, to the power of the generator of the maximal ideal that x is associated to). In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such The case K[[X]] is a special case of the above.
Any field. Define v(x) = 1 for all nonzero x.

The examples of polynomial and power series rings in one variable are the reason that the function v in the definition of a Euclidean domain is not assumed to be defined at 0.

Properties

The following properties of Euclidean domains do not require the inequality v(a) ≤ v(ab):

The extended Euclidean algorithm is applicable (which is the source of the name Euclidean domain). The extended Euclidean algorithm is an extension to the Euclidean algorithm for finding the Greatest common divisor (GCD of integers a and b

Every Euclidean domain is a principal ideal domain. In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i In fact, if I is a nonzero ideal of a Euclidean domain D and a is chosen to minimize v(a) over all nonzero elements of I, then I = aD. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.

The principal ideals of elements with minimal Euclidean valuation are the entire ring, i. e. they are units. In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i (If the inequality v(a) ≤ v(ab) is assumed, all the units have this minimal valuation. )

Every nonzero nonunit is a product of irreducibles. This follows from the corresponding result for any principal ideal domain (or Noetherian domain), though assuming the inequality v(a) ≤ v(ab) would enable a direct inductive argument. In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i In Abstract algebra, a Noetherian ring is a ring that satisfies the Ascending chain condition on ideals.

Conversely, not every PID is Euclidean, though exceptions are not easy to find. For example, for d = -19, -43, -67, -163, the ring of integers of Q( $\sqrt{d}$ ) is a PID which is not Euclidean, but the cases d = -1, -2, -3, -7, -11 are Euclidean. In Mathematics, the ring of integers is the set of Integers made an Algebraic structure Z with the operations of integer addition ^[1]

However, many finite extensions of Q with trivial class group do have Euclidean integral rings. In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. In Mathematics, the extent to which Unique factorization fails in the ring of integers of an Algebraic number field (or more generally any Dedekind domain Assuming the extended Riemann hypothesis, if K is a finite extension of Q and the ring of integers of K is a PID with an infinite number of units, then the ring of integers is Euclidean. The Riemann hypothesis is one of the most important Conjectures in Mathematics. ^[2] In particular this applies to the case of totally real quadratic number fields with trivial class group. In addition (and without assuming ERH), if the field K has trivial class group and unit rank strictly greater than three, then the ring of integers is Euclidean. In Algebraic number theory, Dirichlet 's unit theorem determines the rank of the Group of units in the ring O K ^[3] An immediate corollary of this is that if the class group is trivial and the extension has degree greater than 8 then the ring of integers is necessarily Euclidean.

References

^ Motzkin, Theodore (1949), “The Euclidean algorithm”, Bulletin of the American Mathematical Society 55 (12): 1142-1146, <http://projecteuclid.org/handle/euclid.bams/1183514381>
^ Weinberger, Peter J. (1973), “On Euclidean rings of algebraic integers”, Proceedings of Symposia in Pure Mathematics (AMS) 24: 321-332
^ Harper, Malcolm & Murty, M. Ram (2004), “Euclidean rings of algebraic integers”, Canadian Journal of Mathematics 56 (1): 71-76, <http://www.mast.queensu.ca/~murty/harper-murty.pdf>

Theodore Samuel Motzkin ( 26 March 1908 &ndash 15 December 1970) was an American mathematician Peter Jay Weinberger is a Computer scientist who works at Google Maruti Ram Pedaprolu Murty, FRSC (b 1953 in Guntur, India) is an Indo-Canadian Mathematician, currently head of the Department of

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Contents

Definition

Examples

Properties

See also

References