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Euclidean domain
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In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm applies. Euclidean domains possess many important properties similar to the integers: for example, the fundamental theorem of arithmetic holds in any Euclidean domain. Another important thing to note is that any Euclidean domain is a Bezout domain. In this respect, Euclidean domains play a fundamental role in ring theory; although they are not as strong as fields, one can still prove many important theorems in their context.
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Encyclopedia
In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm applies. Euclidean domains possess many important properties similar to the integers: for example, the fundamental theorem of arithmetic holds in any Euclidean domain. Another important thing to note is that any Euclidean domain is a Bezout domain. In this respect, Euclidean domains play a fundamental role in ring theory; although they are not as strong as fields, one can still prove many important theorems in their context. Note that every field is a Euclidean domain.
A Euclidean domain is a specific type of integral domain. One has the following chain of class inclusions:
Motivation
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).
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.
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. Define v(n) = |n|, the absolute value of n.
- Z[i], the ring of Gaussian integers. Define v(a+bi) = a2+b2, the norm of the Gaussian integer a+bi.
- Z[?] (where ? is a cube root of 1), the ring of Eisenstein integers. Define v(a+b?) = a2-ab+b2, the norm of the Eisenstein integer a+b?.
- K[X], the ring of polynomials over a field K. For each nonzero polynomial f, define v(f) to be the degree of f.
- KX, the ring of formal power series over the field K. 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. 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). The case KX 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):
- Every Euclidean domain is a principal ideal domain. 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.
- The principal ideals of elements with minimal Euclidean valuation are the entire ring, i.e. they are units. (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.
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 is a PID which is not Euclidean, but the cases d = -1, -2, -3, -7, -11 are Euclidean.
However, many finite extensions of Q with trivial class group do have Euclidean integral rings.
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.
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.
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.
See also
- Ordinal number - these allow a kind of Euclidean division: for all a and ß, if ß > 0, then there are unique ? and d such that a = ß · ? + d and d < ß; however, the ordinals are not a Euclidean domain, since they are not even a ring (addition of ordinals, for example, is not commutative).
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